Schur non-positivity conjecture for products of two chains

For positive integers $m$ and $n$, let $\mathbf m\times\mathbf n$ be the product of two chains, and let $\operatorname{inc}(\mathbf m\times\mathbf n)$ denote its incomparability graph. A graph is Schur positive when its chromatic symmetric function is a nonnegative linear combination of Schur functions. Schur non-positivity conjecture. If $n\geq2$ and $m\geq8$, or if $n\geq3$ and $m\geq n+5$, then $\operatorname{inc}(\mathbf m\times\mathbf n)$ is not Schur positive. The paper proves broad families of non-Schur-positive products and gives computational evidence for the remaining ranges covered by this conjecture.