Toric g-vector Kruskal–Katona conjecture for simple polytopes

Let $P$ be a simple polytope, and let $(\gamma_i(P))$ and $(g_i(P))$ denote its $\gamma$-vector entries and toric $g$-vector entries, respectively. Toric g-vector Kruskal–Katona conjecture. If the $\gamma$-vector entries of $P$ satisfy the Kruskal–Katona inequalities, then the toric $g$-vector entries of $P$ satisfy the Kruskal–Katona inequalities as well. This would extend the known relationship between toric $g$-polynomials and face numbers of simplicial complexes beyond the cube, and is suggested by results on $\gamma$-vectors of simple polytopes.