msws32() from the paper is the exact code I wrote above. The "s = 0xb5ad4eceda1ce2a9" is not part of the state, it's the CONSTANT.
I've tested msws32 it passes TestU01s BigCrush and didn't fail in >=1 TB of PractRand (I stopped after that). A scaled down msws16 fails PractRand after 2 GB, a msws24() variant passes >=256 GB (I stopped after that).
It's certainly not as good as more state of the art PRNGs like PCG, xoshiro, romu, sfc64, tylo64, but it is very simple and has quite high quality output, much better than any similarly simple to construct PRNG I know of.
Sorry for misrepresenting it a bit.
The renaming and the having the constant be a variable confused me when skimming for the parts that I was looking for.
So, the state is 128 or 256 for the versions presented and 64 for msws16.
I don't remember if running PractRand in word mode changes the way it reports results but either way failing at 2GB would mean it failed even before going through the whole Weyl sequence although the period itself isn't necessarily reduced.
I'm not sure if the middle-square is acting as a decent non-linear scrambler on the poor adder state or if both combined manage to hold 30 bits worth of state. Swapping the adder with an lcg or lfsr on msws16 would provide an answer.
PractRand has the benefit that we can look at where and how failure happens in these reduced versions so I think the criticism ultimately stands regarding the paper.