The right inner mapping group conjecture for simple right conjugacy closed loops

Let $Q$ be the right conjugacy closed loop $(Q,\circ_f)$ constructed from $GL(2,q)$, where $q=p^n$, and let $\operatorname{Inn}_{\rho}(Q,\circ_f)$ denote its right inner mapping group. For $s\in\mathbb{F}_q$, consider the subgroup of upper triangular matrices in $GL(2,q)$.

$$\operatorname{Inn}_{\rho}(Q,\circ_f)=\left\{\begin{pmatrix}x&y\\0&1\end{pmatrix}\mathrel{\middle|}x=a^2s^m,\ a,y\in\mathbb{F}_q,\ m\in\mathbb{Z}\right\}.$$

The right inner mapping group conjecture. The right inner mapping group of $(Q,\circ_f)$ is exactly

$$\operatorname{Inn}_{\rho}(Q,\circ_f)=\left\{\begin{pmatrix}x&y\\0&1\end{pmatrix}\mathrel{\middle|}x=a^2s^m,\ a,y\in\mathbb{F}_q,\ m\in\mathbb{Z}\right\}.$$

This describes the group sought in the preceding question about $\operatorname{Mlt}_{\rho}(Q,\circ_f)$. The supplied text gives no proof or status information for this assertion, so its resolution remains unclear.