  Let $p$ be a prime number and $F/F^+$ 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 $\overline{r}:
\textrm{Gal}(\overline{F^+}/F^+) \longrightarrow
{}^C\textrm{U}_{1,1}(\overline{\mathbb{F}}_p)$ be a modular $L$-parameter
valued in the $C$-group of a rank 2 unitary group associated to $F/F^+$. We
assume $\overline{r}$ 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
$\overline{\mathbb{F}}_p$-representations of the $p$-adic unitary group
$\textrm{U}_{1,1}(F^+_v)$ associated to $\overline{r}$ in spaces of mod $p$
automorphic forms have Gelfand--Kirillov dimension $[F^+_v:\mathbb{Q}_p]$.
