Chan–Kontorovich–Pak spanning-tree spectrum conjecture

Let $\mathcal{T}(n)$ denote the set of values of the spanning-tree count $\tau(G)$ over unrestricted graphs on $n$ vertices. Cayley's theorem gives $\tau(K_n)=n^{n-2}$, so the possible extremal values are superexponential. Chan–Kontorovich–Pak's conjecture.

$$|\mathcal{T}(n)|=e^{\Omega(n\log n)}.$$

The planar restriction is known to have an exponential-size spectrum, whereas the unrestricted spectrum is conjectured to be superexponential; the source reiterates this conjecture without resolving it.