[tl;dr momentum loss is a generic feature of a spacetime which has a metric expansion of space. Informally, every future direction is uphill, so everything has to roll uphill[*]. If the expansion is accelerated by a cosmological constant the hill steepens with the expansion. The author of the accepted answer at the phys SE link seems to want to argue that the slowdown while rolling uphill is what generates the hill. It's not.
The hill is generated by the initial impulse that kicks off the expansion and driver of the expansion-acceleration if any.]

So, uhhm, let's look at the

> accepted solution in the second link

which was authored by the hagiographic https://www.vixrapedia.org/wiki/Philip_Gibbs (in which there's a whole section "Energy Conservation in General Relativity": "... Gibbs is the leader of a small minority of commentators who dispute [that energy conservation only works in special cases]".

In the comments below the answer he links to a 2010 article in the "Prespacetime Journal" of which he is one of the editors and more prolific contributors. That article, written by Gibbs (which some editing pass managed to leave in a state where one finds both Einstein-Hilbert and Hilbert-Einstein and similar) [ETA: in 1997, see below] finds the right problem -- energy is very hard to define in a general curved spacetime <https://en.wikipedia.org/wiki/Mass_in_general_relativity#Def...>. It does not imho find the right solution. His is ultimately an attempt to define a quasilocal quantity and then extend it to some cosmologically large volume. His method requires a closed cosmology, and he had the decency to say so in the second paragraph. Consequently most people could simply stop reading at that point and spare themselves pp 906-907 where he admits that he's forced into considering coordinate-dependent quantities like everyone else (none of which he cited, more below), and misses the opportunity to back a quasilocal view with an ultralocal (if averaged) one. If he had done that he might have confronted the problem that in the far future of an accelerating expanding spacetime dark energy and horizon radiation are all that's left at a typical point in spacetime.

I agree with mr_mitm that dark energy doesn't make much qualitative difference with respect to energy losses from matter that just don't go anywhere (pace Gibbs). In an inertially expanding non-closed spacetime's far future you find no energy at all at a typical point, so vast vast regions appear to be flat vacuum. If the expansion is accelerated by dark energy, you get vacuum which isn't flat. It's best to consider this by comparing parallel trajectories, Raychaudhuri-style.

Consider two test (massless, non-interacting) particles having an initially-parallel trajectory at an adjacent pair of points in each version. We are setting up the parallel-ness at late times; as we just want to probe the late-time of an expanding universe their earlier configuration is unknown and not important here. These particles will stay parallel in the inertially expanding case, but will diverge in the accelerated expansion (cosmological constant) case.

Both of these cases conflict with Gibbs's stack exchange answer in the following way. Make our test particles massless, and apply the Blau-Frank-Weiss adaptation of Fermi coordinates to their worldlines; the affine parameter x+ along the null geodesics generates a pair of momenta k^\mu = \dot{x+}^\mu and orthogonal "transverse" accelerations a^mu = k^nu \nabla_nu k^mu = 0. The particles lose momentum in their direction of travel but feel no acceleration; they are simply in zero-force constant free fall. The cosmologies themselves pick out solutions of the geodesic equation, and the momentum-loss along null geodesics is because of the expansion of space. The test particles on their worldlines do not know whether the worldlines are parallel or divergent. Importantly, the geodesics are not "driven apart", they just are what they are in the spacetime.

Since we can solve the geodesic equations for the entire spacetime (as a "block universe") it is hard to imagine what one even means by an energy of the gravitational field, unless one slices up the spacetime into time-ordered spaces. But once you do that you have to guard against slicing-dependent quantities like those Gibbs contemplates. He starts, uncontroversially, with a slicing that contains the vast majority of actually-occupied timelike geodesics (the centres of masses of inertially drifting galaxy clusters, represented by the FLRW homogeneous fluids), but does not contemplate other slicings -- do the conclusions hold up for our pair of massless observers? No, his plan only works for for a family of timelike geodesics which were parallel at the termination of the inertial impulse or early in the the accelerated-expansion history. His conclusions also don't hold up in general for slicing along any accelerated massive observer's proper time either, and are super-awkward if we slice along a high-energy cosmic ray propagating from one galaxy cluster to another (e.g. if we wanted to think about its mean free path).

Gibbs's paper, dated 2010 [ETA: turns out to be written in 1997 but published 2010], cites work only up to the early 1990s, with 6/9 of his references dated before 1970. Any peer-review would demand he expand his list of references with the expectation he is or should be aware of work done in the fifteen year gap. I'll quote from the publication's editorial polcies page: "PSTJ employs a three-tier due diligence process in selecting submissions for publication. First, all submissions are screened by the Editor to determine whether the contents and/or forms are suitable for publication in PSTJ; and unsuitable submissions will be declined without reviews. Second, the remaining submissions will be reviewed by the Editor; and submissions passes Editor's review will be accepted for publications". Well, Gibbs is an Editor so maybe this paper didn't take the "Third, the still remaining articles ..." route. Who knows?

Honestly, I expected worse; he only fell into traps that were already in the academic literature and dealt with in grad-level GR textbooks.
<https://en.wikipedia.org/wiki/ViXra> is well-known and "fringe" is polite". That said, this paper was fairly conventionally wrong.

[*] "Uphill" means against a gravitational potential. It is seductive in cosmology to think about gravitational potential and kinetic energy, especially if we drop dark energy. Early in the expanding universe masses like galaxy clusters are all relatively close and flying apart at high speed; at later times the masses are all relatively distant and flying apart at lower speed. The early kinetic energy has been converted into late gravitational potential. (Under time reversal, that gravitational potential energy crunches everything back together at high speed).

If we add dark energy then in the resulting accelerated expansion both the gravitational potential and kinetic energy increase with time. If the dark energy is the cosmological constant (CC), where is the extra kinetic energy coming from? The CC is constant at every place and every time. So there's an awfully big hint that energy isn't conserved, and digging into this one learns that the seductive idea doesn't work.

ETA: darn it, I didn't notice the "Note: this work was completed in January 1997" smooshed in after the author's correspondence address at the bottom of its first page. However, I would think the journal printing it thirteen years later would want to make its age more obvious and/or to publish a revision of the 1997 work. That said, two citations and a self-citation: <https://scholar.google.co.uk/scholar?cites=13647521487929411...>. If you want fringe, check out the mental causation paper there. With the exception in square brackets I added above, I don't think the 1997 vs 2010 date of authorship vs date of publication makes much difference to my comment.