Finite-basis conjecture for grid classes

Let $M$ be a finite matrix whose entries specify the monotone permutation classes placed in the cells of a grid, and let $\operatorname{Grid}(M)$ denote the resulting grid class. A permutation class is finitely based if it can be written as $\operatorname{Av}(B)$ for some finite set $B$ of excluded permutations. Finite-basis conjecture. Every grid class is finitely based. The conjecture is presented in the source as remaining open; the authors note that bases are known for all $2\times2$ matrices but not for larger matrices in general.