Positivity conjecture for the Saxl polynomials

Let $d$ be a positive integer, let $\ol\nu\vdash d$, and let $s_{\ol\nu}$ be the polynomial map associated with the staircase Kronecker-square coefficients. Let $c(\ol\nu)$ and $t(\ol\nu)$ be the interval parameters defined in the paper. Positivity conjecture. The polynomial $s_{\ol\nu}$ is positive on the interval $[t(\ol\nu),\infty)$. This conjecture is stated as equivalent to Saxl's conjecture in the paper and would imply the eventual occurrence of the corresponding irreducible characters in staircase Kronecker squares; the source gives no resolution status.