LQYZ's non-Schur-positivity conjecture for products of chains

Let $\textbf m$ and $\textbf n$ denote chains indexed by the positive integers $m$ and $n$, and let $\textbf m\times\textbf n$ be their product lattice. A lattice is Schur positive when the chromatic symmetric function of its incomparability graph is Schur positive. LQYZ's conjecture. For any positive integer pair $(m,n)$ such that $m\ge 8$ and $n\ge 2$, or that $m\ge n+5$ and $n\ge 3$, the lattice $\textbf m\times\textbf n$ is not Schur positive. This extends a known theorem for a subfamily of products of chains and predicts the full non-Schur-positive range described in the conjecture. The supplied text gives no resolution, so the conjecture remains open.