What is the largest Laver table which has been computed?

On Azimuth, on May 6, 2016, Joseph van Name wrote:

The largest classical Laver table computed is actually $A_{48}$. The 48th table was computed by Dougherty and the algorithm was originally described in Dougherty's paper here. With today's technology I could imagine that one could compute $A_{96}$ if one has access to a sufficiently powerful computer.

One can compute the classical Laver tables up to the 48th table on your computer here at my website.

I've been in contact with Patrick Dehornoy and Ales Drapal and both thought that $A_{28}$ is likely the current record for a Laver table computation.