Mao–Wang's determinant conjecture for walk matrices of rooted products with paths

Let $G$ be a simple graph on $n$ vertices with adjacency matrix $A$, and let $e$ be the all-ones vector. Its walk matrix is

$$W(G)=[e,Ae,\ldots,A^{n-1}e].$$

Let $G\circ P_m$ be the rooted product of $G$ and a path $P_m$, rooted at an endvertex of the path, and let $a_0$ denote the constant term of the characteristic polynomial of $G$.

Mao–Wang's conjecture. For any positive integer $m\ge 2$,

$$\det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m.$$

This conjecture would unify and extend the formulas proved for $m=2$, $m=3$, and $m=4$ by Mao–Liu–Wang and Mao–Wang, respectively. The supplied source gives no resolution of the conjecture.