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 $\bar{\mathbb{F}}_{p}[{\rm GL}_2(\mathbb{F}_{p^2})].$ We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in $\bar{\mathbb{F}}_{p}\otimes det^e$ for some $e \geq 0$; except possibly in some few cases.