Exponential non-isomorphism conjecture for spanning trees of regular graphs

Let $n,d\geq 3$ be integers, and let $G$ be a connected $n$-vertex $d$-regular graph. Write $T_{\mathrm{unlabeled}}(G)$ for the set of isomorphism classes of spanning trees of $G$. Exponential spanning-tree conjecture. There exists a universal constant $c>0$ depending only on $d$ such that

$$|T_{\mathrm{unlabeled}}(G)|\geq e^{cn}.$$

The paper proves an analogous result for sufficiently large degree and asks whether the threshold can be reduced to $d_0=3$; the conjecture asserts this bound for every $d\geq3$.