The rank-4 interlacing-array edge-labeling conjecture

Let $\operatorname{\mathcal{T}}_{4,n}$ be the set of interlacing triangular arrays of rank $4$ and height $n$, and let $\square_n$ be the $n\times n$ square grid graph. An edge labeling of $\square_n$ uses four labels, and it is face-distinct when the four sides of every face have pairwise distinct labels. Rank-4 edge-labeling conjecture. For every $n\geq 0$, $|\operatorname{\mathcal{T}}_{4,n}|$ equals the number of face-distinct edge labelings of $\square_n$.

The conjecture is new in the paper and has been checked through $n=7$; it replaces the refuted rank-4 vertex-coloring conjecture with an edge-labeling enumeration.