Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields

Let $F$ be an imaginary quadratic field and $\mathcal{O}$ its ring of integers. Let $\mathfrak{n}\subset \mathcal{O}$ be a non-zero ideal and let $p>5$ be a rational inert prime in $F$ and coprime with $\mathfrak{n}$. Let $V$ be an irreducible finite dimensional representation of ${\bar{\mathbb{F}}}_p[{\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_{p^2})]\mathrm{.}$ We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\bar{\mathbb{F}}}_p\otimes de{t}^e$ for some $e\ge 0$; except possibly in some few cases.