Conjecture on cross-degree bounds and interlacing for multipartite symmetric edge polytopes

Let $a_1\leq a_2\leq\dots\leq a_k\leq n$ be positive integers. Let $m$ be the cross-degree of the Ehrhart polynomial of the symmetric edge polytope of $K_{a_1,a_2,\dots,a_k}$. For a list of ones, write $1^k$ for $k$ consecutive ones, so that $K_{1^k,n}$ is the corresponding complete multipartite graph.

Cross-degree and interlacing conjecture. The inequalities

$$\left\lfloor\frac{\sum_{i=1}^k a_i}{2}\right\rfloor\leq m+1\leq\sum_{i=1}^k a_i$$

should hold. Furthermore, the Ehrhart polynomial of the symmetric edge polytope of $K_{1^k,n}$ should interlace that of $K_{1^{k+1},n}$.

The conjecture refines the paper's study of cross-degree and interlacing relations for complete multipartite graphs. The supplied context does not state whether these bounds and the proposed interlacing have been proved or remain open.