Sharpness conjecture for characteristic-polynomial congruence bounds of q-Seidel matrices

Let $\mathcal X_n(q,e)$ and $\mathcal X'_n(q,e)$ denote the sets of congruence classes of characteristic polynomials of $q$-Seidel matrices described in Theorems 1 and 2, respectively. For $e\in\mathbb N$ with $e\geqslant 3$ and $q$ a prime power, there exists $N\in\mathbb N$ such that for all $n\geqslant N$ the corresponding upper bounds in those theorems are attained with equality.

Sharpness conjecture. For $e \in \mathbb N$ with $e \geqslant 3$ and $q$ a prime power, there exists $N \in \mathbb N$ such that for all $n \geqslant N$ we have equality in the corresponding bounds in Theorem~ and Theorem~.

The conjecture is motivated by computer experiments for $n=21$ and $e=3,4$, which provide empirical evidence that the bounds are sharp for fixed $q$ and sufficiently large $n$ relative to $e$.