Mazur's tame potential semistability conjecture for classical modular points

Let $\overline{\Omega}$ be the closure of the crystalline locus in the deformation space $X(\overline{\rho})$, and let $\rho \in \overline{\Omega}$ be a classical modular point. Say that a local representation at $p$ is tamely potentially semistable if it becomes semistable after restriction to a tame extension of $\mathbf Q_p$. Mazur's conjecture. The point $\rho$ is tamely potentially semistable at $p$; equivalently, the local representation attached to $\rho$ becomes semistable after restriction to a tame extension of $\mathbf Q_p$. This is stated as one of the two recent conjectures concerning the structure of the closure of the crystalline locus. The source does not provide evidence of a resolution.