Irreducibility conjecture for reductions at the boundary of weight space

Let $f$ be a newform of weight $k\geq 2$ and level $\Gamma_1(Np^r)$, where $r\geq 2$, whose character has $p$-part $\chi$ of conductor $p^r$, and let $\alpha$ be its $U_p$-eigenvalue. Let $V_{k,\alpha,\chi}$ be the associated local Galois representation and let $\overline{V}_{k,\alpha,\chi}$ be the semisimplification of its mod-$p$ reduction. Normalize the valuation $v_\chi$ so that its image on ${\mathbb{Q}}_p(\chi)^\times$ is ${\mathbb{Z}}$.

Boundary irreducibility conjecture. If $v_\chi(\alpha)\notin{\mathbb{Z}}$, then $\overline{V}_{k,\alpha,\chi}$ is irreducible.

This local assertion is proposed to explain patterns in the slopes of forms near the boundary of weight space. The source notes that several global results instead force integral slopes in particular levels, but does not resolve the general conjecture.