Combinatorial invariance of the TNS property for projective toric varieties

Let $X$ and $Y$ be nonsingular projective toric varieties of complex dimension $n$ whose moment polytopes are combinatorially equivalent. Combinatorial invariance conjecture. $X$ is a TNS-manifold if and only if $Y$ is a TNS-manifold. The preceding theorem suggests a connection between the combinatorial type of a smooth projective toric variety's moment polytope and its TNS property. The analogous statement for quasitoric manifolds is generally false in dimensions greater than two, while the conjecture remains open here.