Divisibility conjecture for matching and characteristic polynomials of hypertrees

Let $H\sqsubseteq\mathcal{H}$ be $k$-trees, where $k\geq 3$. Write $\varphi(H)$ for the matching polynomial of $H$ and $\phi(\mathcal{H})$ for its adjacency characteristic polynomial.

Divisibility conjecture. If $H\sqsubseteq\mathcal{H}$ are $k$-trees for $k\geq 3$, then

$$\varphi(H)\mid\phi(\mathcal{H}).$$

In particular, if $H\sqsubseteq\mathcal{H}$, then

$$\phi(H)\mid\phi(\mathcal{H}).$$

The conjecture is motivated by computations for induced subgraphs of a specific 3-uniform hypertree, where each matching polynomial divides the corresponding characteristic polynomial. It asks whether this divisibility persists for all contained $k$-trees and, in particular, for their adjacency characteristic polynomials.