The extremal spanning-tree conjecture for $C_4$-free graphs

Let $q$ be a prime power and let $n=q^2+q+1$. For an $n$-vertex $C_4$-free graph $G$, write $\tau(G)$ for its number of spanning trees, and let $\mathrm{st}(n,C_4)$ denote the maximum of $\tau(G)$ over all such graphs. Let $ER_q$ be the orthogonal polarity graph of the projective plane of order $q$. Extremal spanning-tree conjecture.

$$\mathrm{st}(n,C_4)=\tau(ER_q)=n^{\frac{n-3}{2}},$$

and the maximizers are precisely the orthogonal polarity graphs. This identifies both the maximum number of spanning trees and all extremal graphs in the prime-power case; the supplied text does not establish whether the conjecture is open or resolved.