Bencs–Csikvári conjecture on spanning forests of regular graphs

Let $G$ be a $d$--regular graph on $n$ vertices, and let $F(G)$ denote its number of spanning forests. Bencs–Csikvári conjecture.

$$F(G)\leq \left(\frac{(d-1)^{d-1}}{(d^2-2d-1)^{d/2-1}}\right)^n.$$

This is the proposed sharp analogue for spanning forests of McKay's upper bound for spanning trees; the abstract states that equality in the exponential base would make the right-hand side best possible. The source gives no resolution of the conjecture.