Aggarwal–Borodin–Wheeler's rank-4 interlacing-array coloring conjecture

Let $\operatorname{\mathcal{T}}_{4,n}$ denote the set of interlacing triangular arrays of rank $4$ and height $n$. Let $\square_n$ be the $n\times n$ square grid graph, and let $\boxtimes_n$ be the graph obtained from $\square_n$ by adding both diagonal edges of every face. Aggarwal–Borodin–Wheeler's rank-4 conjecture. For $n\geq 1$,

$$|\operatorname{\mathcal{T}}_{4,n}|=\frac{1}{5}\left|\{\text{proper vertex $5$-colorings of }\boxtimes_n\}\right|.$$

The conjecture is refuted: the paper computes $|\operatorname{\mathcal{T}}_{4,3}|=191232$, whereas the proposed coloring expression gives $187008$.