Finite generation and Cohen–Macaulayness of matrix-valued Hilbert modular forms

Let $r,c\geq 1$ be integers, let $F$ be a totally real field, and let $\rho:\Gamma_F\rightarrow\operatorname{GL}_r(\mathbb{C})$ be a complex representation. The graded space $M^F_c(\rho)$ is considered as a module over the ring $M^F_1(\operatorname{id}_1)$ of scalar Hilbert modular forms. Finite-generation and Cohen–Macaulayness conjecture. $M^F_c(\rho)$ is a finitely generated Cohen–Macaulay module over $M^F_1(\operatorname{id}_1)$. It is known that this holds when $F=\mathbb{Q}$; the conjecture concerns totally real fields in general.