Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2

Let $p$ be a prime number and $F\mathrm{/}{F}^{+}$ a CM extension of a totally real field such that every place of $F^{+}$ above $p$ is unramified and inert in $F$. We fix a finite place $v$ of $F^{+}$ above $p$, and let $\bar{r}:\text{Gal}(\overline{F^{+}}\mathrm{/}{F}^{+})⟶{}^C{\text{U}}_{1,1}({\bar{\mathbb{F}}}_p)$ be a modular $L$-parameter valued in the $C$-group of a rank 2 unitary group associated to $F\mathrm{/}{F}^{+}$. We assume $\bar{r}$ is semisimple and sufficiently generic at $v$. Using recent results of Breuil–Herzig–Hu–Morra–Schraen along with our previous work, we prove that certain admissible smooth ${\bar{\mathbb{F}}}_p$-representations of the $p$-adic unitary group ${\text{U}}_{1,1}(F_v^{+})$ associated to $\bar{r}$ in spaces of mod $p$ automorphic forms have Gelfand–Kirillov dimension $[F_v^{+}:{\mathbb{Q}}_p]$.