Breuil–Buzzard–Emerton irreducibility conjecture for reductions of crystalline representations

Let $p$ be an odd prime, let $k\geqslant 2$ be an integer, and let $a\in\overline{\mathbb{Z}}_p$ satisfy $v_p(a)>0$. Let $V_{k,a}$ be the associated two-dimensional crystalline $p$-adic representation with Hodge–Tate weights $(0,k-1)$, and let $\overline{V}_{k,a}$ denote the semisimplified reduction modulo the maximal ideal of a Galois-stable $\overline{\mathbb{Z}}_p$-lattice in $V_{k,a}$. Breuil–Buzzard–Emerton conjecture. If $k$ is even and $v_p(a)\notin\mathbb{Z}$, then $\overline{V}_{k,a}$ is irreducible. This is a conjecture about reductions of two-dimensional crystalline representations. The paper establishes it for representations over non-subtle components of weight space and for subtle components when the slope is less than $\frac{p-1}{2}$, while the general assertion remains open.