Mazur's principle for U(2,1) Shimura varieties

Let $U$ be a level subgroup with local component $U_{\mathfrak p}=\Gamma_{i_0,\dots,i_r}$, and let $m$ be a maximal ideal of the Hecke algebra associated with $U$. Call $m$ pseudo-Eisenstein if either

$$H^{n-1}_{\text{\rm \tiny 'et}}((X_U)_{\overline{F}_{\mathfrak p}},{\mathbb Z}_l)^{\text{\rm \tiny tor}}_m\ne 0,$$

or $H^i_{\text{\rm \tiny 'et}}((X_U)_{\overline{F}_{\mathfrak p}},{\mathbb Z}_l)_m\ne 0$ for some $i\ne n-1$. Let $\overline{\rho}_m$ be the mod $m$ Galois representation, let $l$ be the residue characteristic of $m$ with $l\ne p$, and let $N$ be the monodromy operator on a representation space for $\overline{\rho}_m$.

Mazur's principle. Suppose that $m$ is not pseudo-Eisenstein and that $N^r=0$ on a representation space for $\overline{\rho}_m$. Then there is a level $U'$ containing $U$, isomorphic to $U$ away from $\mathfrak p$, with

$$U'_{\mathfrak p}=\Gamma_{i'_0,\dots,i'_{r'}},\qquad r'<r,$$

such that $\overline{\rho}_m$ arises from level $U'$.

This is a converse to the necessary monodromy-vanishing condition for lowering the level at $\mathfrak p$, extending the role of Mazur's principle from modular curves to these unitary Shimura varieties. The supplied text does not establish the conjecture or state results proving it, so its resolution status is left open.