  Let $F$ be a totally real field in which $p$ is unramified. Let
$\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a
modular Galois representation which satisfies the Taylor-Wiles hypotheses and
is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding
Hecke eigensystem. Then the $\mathfrak{m}$-torsion in the mod $p$ cohomology of
Shimura curves with full congruence level at $v$ coincides with the
$\mathrm{GL}_2(k_v)$-representation $D_0(\overline{r}|_{G_{F_v}})$ constructed
by Breuil and Pa\v{s}k\={u}nas. In particular, it depends only on the local
representation $\overline{r}|_{G_{F_v}}$, and its Jordan-H\"older factors
appear with multiplicity one. This builds on and extends work of the author
with Morra and Schraen and independently of Hu-Wang, which proved these results
when $\overline{r}|_{G_{F_v}}$ was additionally assumed to be tamely ramified.
The main new tool is a method for computing Taylor-Wiles patched modules of
integral projective envelopes using multitype tamely potentially Barsotti-Tate
deformation rings and their intersection theory.
