Saturated Newton polytope conjecture for Kronecker products of Schur functions

For a symmetric function $f$, let $f(x_1,\ldots,x_k)$ denote its specialization obtained by setting $x_m=0$ for all $m\geq k+1$. A polynomial $f(x_1,\ldots,x_k)=\sum_{\alpha}c_\alpha x^\alpha$ has a saturated Newton polytope (SNP) when the points with positive coefficients coincide with the lattice points in their convex hull. For partitions $\lambda$ and $\mu$, write

$$s_\lambda*s_\mu=\sum_\nu g(\lambda,\mu,\nu)s_\nu,$$

where $g(\lambda,\mu,\nu)$ are the Kronecker coefficients. Kronecker SNP conjecture. The Kronecker product $s_\lambda*s_\mu$ has a saturated Newton polytope. This conjecture asks for a saturated Newton polytope property for one of the central products in algebraic combinatorics and representation theory. The supplied text gives no resolution evidence for this claim.