The size of the spanning-tree spectrum of simple graphs

For a graph $G$, let $\tau(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $\tau(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for all sufficiently large $n$. This is optimal up to the choice of the constant $c$ and resolves a conjecture of Chan-Kontorovich-Pak regarding a problem of Sedl\'a\v{c}ek from the late 1960s.