Higashitani–Kummer–Michałek conjecture for complete multipartite graphs

Let $K_{a_1,\dots,a_k}$ be a complete multipartite graph, and let $E_{a_1,\dots,a_k}$ denote the Ehrhart polynomial of its symmetric edge polytope. Let

$$\operatorname{CL}=\{z\in\operatorname{\mathbb{C}}\colon\Re(z)=-\frac{1}{2}\}$$

be the canonical line. Two real-rooted polynomials interlace when their roots alternate on a common totally ordered set.

Higashitani–Kummer–Michałek conjecture. (i) For any complete multipartite graph $K_{a_1,\dots,a_k}$, the roots of $E_{a_1,\dots,a_k}$ lie on $\operatorname{CL}$. (ii) If $a_1\leq\dots\leq a_k$, then $E_{a_1,\dots,a_k}$ and $E_{a_1,a_2,\dots,a_k-1}$ interlace on $\operatorname{CL}$.

This extends the known result for symmetric edge polytopes of $K_{1,n}$, $K_{2,n}$, and $K_{3,n}$, and would give a broad class of $\operatorname{CL}$-polytopes. The conjecture remains open in the general complete multipartite case.