Lee's polynomial anticoncentration conjecture for random spanning trees

Let $d$ be sufficiently large, let $G$ be a connected graph with $n$ vertices and minimum degree at least $d$, and let $\mathcal{T}$ be a uniformly random spanning tree of $G$.

Lee's conjecture. For every tree $T$,

$$\mathbb{P}(\mathcal{T} \cong T) \le n^{-\Omega(1)}.$$

This conjecture seeks a polynomial anticoncentration bound in graphs of large minimum degree, generalizing the exponential anticoncentration known for almost regular graphs. The complete bipartite graph $K_{d,n-d}$ shows that an exponential bound in $n$ is impossible in this setting.