Hall's matching-polynomial conjecture for cycle covers

Let $C_n$ be the cycle graph on $n$ vertices, let $P_m$ be the path graph on $m$ vertices, and let $d4d0_{G,d}(x)$ denote the $d$-matching polynomial of a graph $G$, defined as the average of the matching polynomials of its $d$-covers. Hall's conjecture. For the cycle graph $C_n$,

$$\mathcal{M}_{C_n,d}(x)=\frac{\mathcal{M}_{P_{nd+n-1}}(x)}{\mathcal{M}_{P_{n-1}}(x)}.$$

This conjecture gives an explicit expression for the $d$-matching polynomial of a cycle in terms of the classical matching polynomials of paths. The surrounding discussion identifies it as a conjecture made independently by Hall; no resolution is supplied here.