Saturated Newton polytopes for sums of Schur polynomials

Prove or disprove that, for every symmetric polytope P and the polynomial s formed by summing Schur polynomials indexed by the pairwise dominance-maximal partition lattice points of P, the polynomial t s formed from all multisets of t such partitions satisfies Newt(t s) = t Newt(s) and has a saturated Newton polytope for every positive integer t.

Background

For a symmetric polytope P, the paper defines a polynomial s as the sum of Schur polynomials indexed by the dominance-maximal partition lattice points of P, so that Newt(s) = P. More generally, for partitions λ1, …, λk, the paper defines t s by summing Schur polynomials indexed by all sums of t partitions selected with repetition from λ1, …, λk.

The conjecture asserts both that the Newton polytope of this constructed polynomial is the t-fold dilation of the Newton polytope of s and that the polynomial contains every lattice exponent in that Newton polytope. If true, this would provide the key input for proving IDP for the associated symmetric polytopes through the tableau-decomposition framework developed in Theorem 3.5. The paper verifies the resulting claim for the 2-partition maximal polytopes in the treated three-dimensional hyperplane case, but states the general assertion as a conjecture.