Nonexistence conjecture for {k, k}-equivelar wnp maps

Let $K$ be a wnp map, meaning a polyhedral map attaining equality in the lower bound $n\geq (k-1)^2$ for an $n$-vertex $\{k,k\}$-equivelar polyhedral map. Nonexistence conjecture. There does not exist any $\{k,k\}$-equivelar wnp map for $k\geq 7$. This would extend the known nonexistence for $k=6$, where no 25-vertex $\{6,6\}$-equivelar polyhedral map exists, and would show that the lower bound is never attained for equal face and vertex degrees at least seven.