In my PhD, I measured the proton form factor and, from that, calculated the radius. So let me explain my view :)

There are essentially three ways to measure the radius:

1) Electron-proton scattering. This gives you the form factors (related to the charge distribution by the Fourier transform), and from that you can calculate the proton radius.
2) Measurements of the electron energy levels of "normal" Hydrogen
3) Measurements of the muon-proton energy levels.

Re 1)
The Mainz proton form factor experiment is to date the most precise proton radius determination from electron scattering and is compatible with the larger value. Our results are along the line of earlier measurements using similar techniques. The first measurements of this kind where done in the 50's and 60's (but produced quite a wrong radius).

Re 2) These measurements are very hard, the proton size effect is very small. Nevertheless, the results are of similar precision as those from 1) and give a compatible radius

Re 3) In muonic hydrogen, the muon is "much closer" to the proton and therefore the proton size has a much larger influence. Because of this, the method produces by far the best precision.

The 7 standard deviations are calculated with the precision of 1) and 2), the error of 3) is negligble!

So, what can go wrong?

In my opinion, the muon experiment is very clean, so I don't believe in an experimental error. The fact that 1) and 2) agree make it unlikely that they are wrong, as they are complete separate methods. However, it is possible.

It could be that we are missing an important part in our understanding of the radiative corrections, i.e. the theory needed to calculate the levels.
This could mean a simple error in one of the calculations, but most if not all of them have been checked by different groups. It could also mean a flaw in one of the solving techniques. Or maybe something which was overlooked.

It is also possible that there is another particle at work here. A possible candidate is a dark photon. This solution has some benefits, especially since such a new particle might also solve the muon g-2 puzzle. But it is not easy to construct a theory of such a particle without violating other experiments. A lot of fine tuning.

It could also be that the muon just behaves differently from the electron. That would shatter a rather basic and widely accepted believe.

I attended a very interesting workshop recently which focused on this puzzle. Unfortunately, we didn't find a solution. However, there are a lot of experiments in the pipeline which might clarify the situation:
- There are several experiments to measure the proton radius using eletron scattering, with specialized instruments and new methods.
-Muse is an experiment which will scatter a combined electron and muon beam from protons. This will test certain aspects of the radiative corrections and allow a direct comparison of the exctracted radii.
-There will be measurements of other muonic atoms.

All in all, this is a very interesting topic right now, with the added benefit that it brings together the often separate communities of nuclear and atomic physics.

Nice tidbit: Our result and the first muonic result was presented at the same conference... and both speakers didn't know what the other would say.

Other nice tidbit: The first try at the muonic measurement didn't work out. When they got it working, they scanned the energy region around the suspected radius value. But they didn't find a resonance. After improving the experiment and double checking everything, they still didn't find anything, until they started looking further away, when suddenly, the resonance appeared. They just didn't scan wide enough the first time!