Irreducible $p$-modular representations of unramified $U(2,1)$

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $\pi$ of $G$ over $\overline{\mathbf{F}}_p$, with an underlying irreducible smooth representation $\sigma$ of a maximal compact open subgroup $K$, we prove that $\pi$ admits eigenvectors for an appropriate Hecke operator $T_\sigma$, and we classify those $\pi$ with non-zero eigenvalues for $T_\sigma$ by a tree argument; as a corollary, we show $\pi$ is supersingular if and only if it is supercuspidal.