tl;dr: gravitational waves do hard-to-calculate things inside a strong ergoregion around a fast-spinning black hole, and may do hard-to-calculate things to the black hole, and those calculations are outside the scope of the paper.

> [how to define] 'slowly rotating' [might be] pretty cool

It's for a pretty cool reason. Bear in mind that I am not a superhuman so have not read the 900-page document rather than scanning through the most interesting bits. Also, forgive me, I got a bit lazy and have left in a bunch of redundancy below rather than trimming it out or reorganizing it.

The question in a nutshell is to whether any setup of initial conditions that are reasonably similar to a Kerr spacetime will settle into another set of conditions reasonably similar to Kerr spacetime eventually. The conditions are roughly (eternal!) Kerr black hole in the distant distant past plus a bunch of gravitational radiation to it's "left" travelling rightwards. The "present" is the collision of the waves and the black hole. The far far future is the remnants of the gravitational waves waaay to the black hole's right, plus the original black hole (albeit with a slightly different spin or mass). There is never anything but gravitational waves and the black hole. How do we know from the mathematics of General Relativity that the far-future black hole still spins at all? Or that it hasn't self-destructed? Does this hold up even as we add more gravitational waves coming from different directions? That's what the authors set out to show.

Lumpy, slowly rotating noncompact (i.e. entirely outside of their Kerr horizons) self-gravitating objects see their surfaces smooth out over time through various processes. Stars find themselves in hydrostatic equilibrium even as things fall into them. Planets are defined as being in hydrostatic equilibrium: their lumpiness and tumbling fades away and tends towards being nearly spherical and spinning around a single axis, even though rocks or ice might fall onto them from time to time.

<https://en.wikipedia.org/wiki/Hydrostatic_equilibrium#Astrop...>
<https://en.wikipedia.org/wiki/List_of_gravitationally_rounde...>

A "peaceful" black hole horizon is highly analogous to the surface of a body in hydrostatic equilibrium. There are some differences here: we can throw "too much" at a rocky planet or star, totally disrupting them. You can't break apart a black hole in the setting under consideration (and likely not at all). You can "bounce" a rock with a glancing blow off a rocky planet, maybe turning the planet into something like Earth-Moon. Again, you can't split a black hole, and you can't bounce something off a black hole's event horizon. You can throw too much at a star and create a supernova. You can't explode a black hole that way. So we want to restrict our analogizing to the case where a relatively small body lands on (but does not destroy or blow chunks out of) a rocky body or star.

Also, while we can spin a star or planet so fast that it disintegrates, we can't do that to a black hole. A glancing blow interaction might speed up the spin of a star or planet, possibly making it spin so fast it rips apart. Spinning a black hole as fast as you can does weird things in the immediate neighbourhood of a black hole, but should not rip the black hole apart. The paper sets out to prove that.

The Kerr stability conjecture that is central to this paper considers the case where the Kerr black hole is perturbed by gravitational radiation "thrown" at it. The black hole relaxes back into Kerr (this is "stability for Cauchy data"). The paper also considers the case where if you are sufficiently far from the perturbed black hole you only see Kerr anyway when light-speed news of the event catches up to your gravitational-wave observatory (this is "stability for scattering data")).

More colloquially, if you start with a slowly-spinning black hole and a (relatively distant) "mess" of gravitational waves, do you eventually end up with a slowly-spinning black hole (or something very close to it) or do you destroy your black hole?

Going from "of course the remains of this sort of interaction must include a spinning black hole, because linearizations of General Relativity show that to be the case for black hole mergers" ("smash two black holes together, you get one bigger black hole and a bunch of gravitational radiation and we can see all of this at LIGO/VIRGO and other telescopes") to "here's a rigorous mathematical description in the full non-linearized theory of General Relativity that is good up to arbitrarily large incoming gravitational radiation: you always get a bigger (slowly spinning) black hole" is roughly the subject of the long paper discussed at the Quanta link and found at <https://arxiv.org/abs/2205.14808>.

The size of a Kerr ergosphere -- the region outside the outer event horizon where things cannot remain still relative to the distant universe -- is determined by a combination of the black hole's mass and its spin angular momentum. If the mass/spin ratio is very high, then mass effectively determines the ergoregion's volume. A low spin also means the variation from the pole (where the ergo-effects are zero) to the equator (where they are strongest) is small.

With a low mass/spin ratio (fast spin), the larger ergoregion can "hold" more gravitational radiation, and the strong ergo-effect leads to a stronger interaction between the Kerr black hole and any gravitational wave inside its ergoregion. This is even harder to calculate than the hardest equations in the paper. It is conceivable that the "held" gravitational radiation could be concentrated while within the ergoregion, and that if the ergoregion were large and strong enough, and the incoming gravitational radiation were "just so", a second black hole could form in the ergoregion from the collapse of the concentrated gravitational waves. In which case [a] would it be flung away to the far reaches of the universe [b] crash into the Kerr black hole merger-style or [c] hang around in a stable orbit near the black hole? [a] and [b], no problem, we have an "asymptotic" Kerr spacetime. [c]: big problem, because the spacetime is now like a barbell, and that sheds gravitational waves. (A single rotating black hole does not shed gravitational waves).

These authors don't seem to say this outright, but they do cite an abundance of papers where the reasons proofs are harder as mass/spin approaches 1:1 are set out by their respective authors.