The Breuil–Mézard cycle-map conjecture for varying determinant

Let $K$ be a non-archimedean local field of residue characteristic $l$, let $\mathbb{F}$ be a finite field of characteristic $p\ne l$, and let $\overline\rho:G_K\rightarrow\operatorname{GL}_n(\mathbb{F})$ be continuous. Put $\mathtt{K}=\operatorname{GL}_n(\mathcal{O}_K)$, and let $\mathcal{Z}$ denote the cycle group of the relevant universal deformation ring. Breuil–Mézard cycle-map conjecture. There exists a mod-$\lambda$ cycle map

$$\overline{\operatorname{cyc}}:\operatorname{R}_{\mathbb{F}}(\mathtt{K})\longrightarrow\mathcal{Z}(\operatorname{R}_{\overline\rho}^{\square}/\lambda)$$

compatible with reduction and the component map; moreover, for every character $\chi$ lifting $\det\overline\rho$, there is a fixed-determinant map

$$\overline{\operatorname{cyc}}^{\chi}:\operatorname{R}_{\mathbb{F}}(\mathtt{K})\longrightarrow\mathcal{Z}(\operatorname{R}_{\overline\rho}^{\square,\chi}/\lambda)$$

fitting into the analogous commutative diagram. This is the full cycle-map formulation of Breuil–Mézard, combining the unrestricted and fixed-determinant versions. It packages the relation between mod-$p$ types and deformation-ring components; the source presents it as the principal conjectural formulation for $l\ne p$.