Interlacing conjecture for Ehrhart polynomials of complete multipartite graphs

Let $G$ be a complete $k$-partite graph of type $(a_1,\dots,a_k)$, and let $H_{a_1,\dots,a_k}$ denote its Ehrhart polynomial. Complete multipartite Ehrhart conjecture. The polynomial $H_{a_1,\dots,a_k}$ has only real roots. Moreover, if $a_1\geq\dots\geq a_k$, then $H_{a_1,\dots,a_k}$ and $H_{a_1-1,a_2,\dots,a_k}$ are $R$-interlaced.