Irreducibility conjecture for reductions of crystalline representations

Let $p$ be odd, let $k$ be even, and let $a_p\in\overline{{\mathbb{Q}}}_p$ satisfy $v(a_p)>0$. Let $V_{k,a_p}$ be the two-dimensional crystalline representation with Hodge–Tate weights $0,k-1$ whose crystalline Frobenius has characteristic polynomial $X^2-a_pX+p^{k-1}$, and let $\overline{V}_{k,a_p}$ denote its reduction modulo $p$.

Irreducibility conjecture. If $v(a_p)\notin{\mathbb{Z}}$, then $\overline{V}_{k,a_p}$ is irreducible.

This local conjecture is motivated by the relationship between reductions of crystalline representations and regularity questions for modular forms. The source presents it as unproved, while noting that the relevant global results concern cases in which local reductions are reducible for other reasons.