Polynomial anticoncentration conjecture under a minimum-degree condition

Let $d_0,c>0$ be universal constants. For every integer $d\geq d_0$, let $G$ be a connected $n$-vertex graph with minimum degree at least $d$, and let $\mathcal{T}$ be a uniformly random spanning tree of $G$. For every $n$-vertex tree $T$, Minimum-degree anticoncentration conjecture.

$$\Pr\bigl[\mathcal{T}\simeq_{\mathrm{iso}}T\bigr]\leq n^{-c}.$$

This proposes a polynomial anticoncentration bound under the minimum-degree assumption, which is substantially weaker than almost regularity. Complete bipartite graphs show that the paper's stronger exponential conclusions fail under this weaker hypothesis, but the authors note that such graphs still contain polynomially many pairwise non-isomorphic spanning trees.