Ulmer's semisimplicity conjecture for the Hecke operator at level Np

Let $p$ be a prime, let $N$ be coprime to $p$, and let $S_k(\Gamma_0(Np),\mathbb Q_p)$ be the space of cusp forms of weight $k$ and level $\Gamma_0(Np)$. For a normalized cuspidal eigenform $f\in S_k(N,\mathbb C_p)$, write $a_p$ for the eigenvalue of the Hecke operator $\mathrm{T}_p$ acting on $f$.

Ulmer's conjecture. The action of $U_p$ on $S_k(\Gamma_0(Np),\mathbb Q_p)$ is semisimple. In particular, the polynomial

$$x^2-a_p x+p^{k-1}$$

always has distinct roots.

This conjecture concerns the diagonalizability of the $U_p$ operator on oldforms and would ensure that the two associated $U_p$-eigenforms have well-defined, separately determined slopes. The source gives no resolution, so the conjecture is recorded as open.