The SNP conjecture for sums of Schur polynomials of symmetric polytopes

Let $\mathcal{P}$ be a symmetric polytope, let $\lambda_1,\lambda_2,\dots,\lambda_k$ be partitions, and set $s:=s_{\lambda_1}+s_{\lambda_2}+\cdots+s_{\lambda_k}$. For $t\in\mathbb Z_{>0}$, let $\multiset{\{1,2,\dots,k\}}{t}$ denote the multisets of size $t$ drawn from $\{1,2,\dots,k\}$, define $\lambda_I:=\sum_{i\in I}\lambda_i$, and set

$$ts:=\sum_{I\in\multiset{\{1,2,\dots,k\}}{t}}s_{\lambda_I}.$$

SNP conjecture. For $t\in\mathbb Z_{>0}$, we have

$$\operatorname{Newt}(ts)=t\operatorname{Newt}(s)$$

and thus $t\mathcal{P}=\operatorname{Newt}(ts)$. Furthermore, $ts$ has saturated Newton polytope (SNP) for all $t$.

This conjecture proposes that the specified sums of Schur polynomials simultaneously realize every dilate of the associated symmetric polytope as a Newton polytope and have SNP, a property used to establish the integer decomposition property. The supplied text gives no resolution, so the conjecture remains open.