Universal determinant formula conjecture for adjacency walk matrices of rooted-product preservers

Let $H^{(v)}$ be an $\mathcal{F}$-preserver of order $m$, and let $G$ be any graph. Write $A(G)$ for the adjacency matrix of $G$, $W_A(G)$ for its adjacency walk matrix, and $G\circ H^{(v)}$ for the rooted product. Universal determinant formula conjecture. One has

$$\det W_A(G\circ H^{(v)})=\pm (\det A(G))^{\left\lfloor\frac{m}{2}\right\rfloor}(\det W_A(G))^m.$$

This conjecture is motivated by an exhaustive search for $\mathcal{F}$-preservers of order at most $10$, where all listed preservers exhibited this form. A general proof is not supplied and remains open.