Clark–Cooper divisibility conjecture for matching polynomials of uniform trees

Let $\mathcal{H}$ be an $r$-tree with $r\geq 3$, and let $H$ be a subgraph induced by some vertex subset. Let $\varphi_H$ denote the matching polynomial of $H$, and let $\phi_{\mathcal{H}}$ and $\phi_H$ denote their characteristic polynomials. Clark–Cooper divisibility conjecture. One has

$$\varphi_H\mid\phi_{\mathcal{H}}$$

and, in particular,

$$\phi_H\mid\phi_{\mathcal{H}}.$$

This conjecture extends the known divisibility of the matching polynomial of a hypertree by its characteristic polynomial. The supplied text attributes it to Clark and Cooper; its resolution is not given here.