The geometric griddability and finite-letter-graph conjecture

Let $X$ be a class of permutations, and let ${\cal G}_X$ be the corresponding class of permutation graphs. A permutation class is geometrically griddable when it can be represented by a geometric grid class.

Geometric griddability and finite-letter-graph conjecture. $X$ is geometrically griddable if and only if ${\cal G}_X$ is a class of $k$-letter graphs for some finite value of $k$.

This conjecture proposes that geometric grid classes of permutations and finite-letter graphs describe the same underlying structure through permutation graphs. The paper establishes constructive polynomial-time recognition for $3$-letter graphs, but the conjectured equivalence is not stated as resolved here.