Serre weight conjectures for $p$-adic unitary groups of rank 2

We prove a version of the weight part of Serre’s conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F\mathrm{/}{F}^{+}$ denote a CM extension of a totally real field such that every place of $F^{+}$ above $p$ is unramified and inert in $F$, and let $\overline{r}: $\text{Gal}$(\overline{F^+}/F^+) \longrightarrow {}^C$\mathbf{U}$_2(\overline{$\mathbb{F}$}_p)$ be a Galois parameter valued in the $C$-group of a rank 2 unitary group attached to $F\mathrm{/}{F}^{+}$. We assume that $\bar{r}$ is semisimple and sufficiently generic at all places above $p$. Using base change techniques and (a strengthened version of) the Taylor-Wiles-Kisin conditions, we prove that the set of Serre weights in which $\bar{r}$ is modular agrees with the set of Serre weights predicted by Gee-Herzig-Savitt.