Exponential anticoncentration conjecture for random spanning trees

Let $\varepsilon>0$ be real. There is a constant $d_{\varepsilon}>0$ such that for every integer $d\geq d_{\varepsilon}$, every connected $n$-vertex $d$-regular graph $G$, and a uniformly random spanning tree $\mathcal{T}$ of $G$, let $T$ be any $n$-vertex tree. Optimal anticoncentration conjecture.

$$\Pr\bigl[\mathcal{T}\simeq_{\mathrm{iso}}T\bigr]\leq e^{-(1-\varepsilon)n}.$$

The conjecture is motivated by the expected number of copies of a fixed tree in a random graph and is supported for the path by the Brégman–Minc inequality. The stated exponential scale is best possible up to lower-order terms, while extension to suitable $(n,d,\lambda)$-graphs is posed as an additional direction.