I don't think so. If you follow the logic of the article to eliminate 6, you essentially have to turn everything too big into integers mod 6. I guess you could also no longer have rational or real numbers or only really limited versions of them without access to all the integers. Losing the real numbers kills classical physics.

It seems space could be at most a 6 x 6 x 6 lattice, maybe forming a hyper-torus or something like that. And time, only 6 possible points in time, too. Hopefully nobody starts walking around in that tiny universe and discovers that there are more than 5 distinct places to visit. To really get rid of 6, you probably have to limit the entire universe to 6 states, but even then I am not really confident that 6 remains out of reach.

Note that there is also a tension between ordinals and cardinals, you can number points in each dimension of the lattice universe with 0 to 5 but that are of course 6 numbers. But I decided to ignore that, it just makes the idea even harder to realize.