Nonpolynomiality conjecture for semi-magic hypercube counting functions

Let $H_n^d$ denote the counting function for semi-magic $d$-dimensional hypercubes of order $n$. Nonpolynomiality conjecture. The function $H_n^d$ is not a polynomial when $n,d\geq 3$. The claim is posed as a challenge to the reader and is motivated by the cited result that $H_3^3$ is a quasi-polynomial but not a polynomial; its general validity remains open.