The exceptional-pair conjecture for quadratic forms over finite fields

Let $\mathbb{F}_{q^m}$ be a finite field extension of $\mathbb{F}_q$, with $m=3$ or $m=4$. The pairs $(3,3)$, $(5,3)$, $(6,3)$, and $(3,4)$ are the only possible exceptional pairs for the existence problem studied by the algorithm.

Exceptional-pair conjecture. For the fields $\mathbb{F}_{q^m}$ with $m=3$ or $m=4$, the only possible exceptional pairs $(q,m)$ are

$$(3,3),\ (5,3),\ (6,3),\ (3,4).$$

This claim is presented as the outcome of sufficient computation using the algorithm, but the supplied text does not establish whether these candidate exceptional pairs are genuinely exceptional or whether the assertion has been proved. Its resolution therefore remains unclear from the source.