  Let $F$ be a totally real field in which $p$ is unramified and $B$ be a
quaternion algebra which splits at at most one infinite place. Let
$\overline{r}:\mathrm{Gal}(\overline{F}/F)\to
\mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation
which satisfies the Taylor-Wiles hypotheses. Assume that for some fixed place
$v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $ \mathbb{Q}_p$ and
$\overline{r}$ is generic at $v$. We prove that the admissible smooth
representations of the quaternion algebra over $\mathbb{Q}_p$ coming from mod
$p$ cohomology of Shimura varieties associated to $B$ have Gelfand-Kirillov
dimension $1$. As an application we prove that the degree two Scholze's functor
vanishes on supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We
also prove some finer structure theorem about the image of Scholze's functor in
the reducible case.
