Local constancy conjecture for crystalline lifts

Let

$$\bar{r}:{\operatorname{Gal}}(\overline{{\mathbb{Q}}}_p/{\mathbb{Q}}_p)\to\operatorname{GL}_2(\overline{{\mathbb{F}}}_p)$$

be a reducible representation. Let $n\geq 0$ be an integer, and let $k,k'\geq n+1$ satisfy $k\equiv k'\pmod{(p-1)p^{M(n)}}$.

Local constancy conjecture. There is a crystalline lift of $\bar r$ with Hodge–Tate weights $0,k-1$ and slope $n$ if and only if there is a crystalline lift of $\bar r$ with Hodge–Tate weights $0,k'-1$ and slope $n$.

This is a purely local analogue of the Gouvêa–Mazur slope-local-constancy conjecture. The source proposes it as a possible route to controlling slopes up to multiplicity, but gives no resolution.