Breuil–Herzig global ordinary-part conjecture for indecomposable points

Let $X$ be the parameter space appearing in the global setting, let $L$ be the coefficient field, and let $\Sigma_p$ be the set of places above $p$. For a generic ordinary point $z=(\chi_z,\lambda_z)\in X(\overline{\mathbf Q}_p)$ with $\rho_{z,v}$ indecomposable for every $v\in\Sigma_p$, define

$$\Pi(\rho_{z,p})^{\operatorname{ord}}=\widehat{\bigotimes}_{v\in\Sigma_p}\Pi(\rho_{z,v})^{\operatorname{ord}}.$$

Let $\widehat H^0(K^p)^{\lambda_z,\operatorname{ord}}_L$ denote the ordinary part of the $G_{\Sigma_p}$-representation $\widehat H^0(K^p)^{\lambda_z}_L$. Breuil–Herzig's global ordinary-part conjecture. Suppose that $z\in X(L)$ is a generic ordinary point and that $\rho_{z,v}$ is indecomposable at each place $v\in\Sigma_p$. Then there exists an integer $d\geq 1$ and a $G_{\Sigma_p}$-equivariant isomorphism

$$\left(\Pi(\rho_{z,p})^{\operatorname{ord}}\right)^{\oplus d}\simeq \widehat H^0(K^p)^{\lambda_z,\operatorname{ord}}_L.$$

This is a global realization conjecture of Breuil and Herzig: the ordinary local representations should occur, with finite multiplicity, in the ordinary completed cohomology. The paper restricts the conjecture to the indecomposable case, while the more general generic ordinary formulation is attributed to Breuil and Herzig.