Alternating-sign conjecture for the Saxl polynomials

Let $d$ be a positive integer, let $\ol\nu\vdash d$, and let $s_{\ol\nu}(x)$ be the polynomial map associated with the staircase Kronecker-square coefficients. Let $c(\ol\nu)$ be the interval parameter defined in the paper. Alternating-sign conjecture. The polynomial $s_{\ol\nu}$ has nonzero integer coefficients on the interval $[c(\ol\nu),\infty)$, and its signs alternate: for every $0\leq k\leq d$, the coefficient of $x^k$ in $s_{\ol\nu}(x)$ has sign $(-1)^{d-k}$. This conjecture concerns the coefficient structure underlying the positivity results and is presented as the second conjecture of the paper; the source gives no resolution status.