The exceptional-group count conjecture for groups of order $p^6$

Let $p\geq 5$ be prime, and let an exceptional group be a finite group $G$ having a normal subgroup $N$ such that $\mu(G/N)>\mu(G)$, where $\mu(G)$ is the least integer $n$ for which $G$ embeds in the symmetric group $S_n$. The groups under consideration have order $p^6$. Exceptional-group count conjecture. The number of exceptional groups of order $p^6$ for primes $p\geq 5$ is

$$\frac{11p+107}{2}.$$

The authors identify this many groups and prove that the proportion of exceptional groups of order $p^6$ tends asymptotically to $0$ as $p$ grows. Computations for $5\leq p\leq 13$ give precisely this value, but the absence of further exceptional groups for all primes $p\geq 5$ remains conjectural.