The order conjecture for strong Fermat solutions at p equals 3

Let $F_m$ be the field appearing in the source. For $m>0$ and a prime $q$ satisfying the stated congruence conditions and totally split in $F_m/\mathbb{Q}$, consider solutions $(u,v)$ of the strong Fermat last theorem equation for $p=3$.

Order conjecture. There exists such a solution $(u,v)$ for which the order of $v/u$ modulo $q$ is at least $m$.

This conjecture is motivated by computations for $p=3$ and concerns the existence of strong Fermat solutions with prescribed large multiplicative order; no resolution is supplied.