A sharp upper bound on the number of spanning forests of regular graphs

Let $G$ be a simple graph on $n$ vertices, and let $F(G)$ denote the number of its spanning forests. Bencs and Csikvári [Upper bound for the number of spanning forests of regular graphs, European J. Combin. 110 (2023) 103677] proved that every $r$-regular graph $G$ with $r\ge 2$ satisfies $F(G)\le r^n$. They further conjectured that for $r\ge 3$, $$F(G{)}^{1\mathrm{/}n}\le \frac{(r-1{)}^{r-1}}{(r^2-2r-1{)}^{r\mathrm{/}2-1}}\mathrm{.}$$ In this paper, we resolve this conjecture in the affirmative.