The polynomial lower-bound conjecture for non-isomorphic spanning trees

Let $d$ be sufficiently large, and let $n$ be sufficiently large relative to $d$. Suppose that $G$ is a connected graph with $n$ vertices and minimum degree at least $d$.

Non-isomorphic spanning-tree lower-bound conjecture. The number of non-isomorphic spanning trees of $G$ is at least

$$\Omega(n^{d-1}).$$

The complete bipartite graph $K_{d,n-d}$ has $\Omega(n^{d-1})$ non-isomorphic spanning trees, motivating this as an essentially optimal lower bound. The preceding sharp anticoncentration conjecture would imply only the weaker lower bound $n^{(1/2-o_n(1))(d-1)}$, so the stronger claim remains open.