Let $N \ge 1$ and let $\ell$ and $p$ be primes not dividing $N$.

The classical Ihara lemma says that if $Y_1(N, \ell)$ is the modular curve attached to the subgroup $\Gamma_1(N) \cap \Gamma_0(\ell)$, and $\operatorname{Pr}_1$ and $\operatorname{Pr}_2$ are the two pushforward maps
$$ H^1(Y_1(N, \ell), \mathbf{Z}_p) \to H^1(Y_1(N), \mathbf{Z}_p) $$
corresponding to $z \mapsto z$ and $z \mapsto \ell z$ on the upper half-plane, then the map $\operatorname{Pr}_1 \oplus \operatorname{Pr}_2$ is surjective onto $H^1(Y_1(N), \mathbf{Z}_p)^{\oplus 2}$, after localising at a non-Eisenstein maximal ideal of the Hecke algebra.

I'm interested in whether this continues to hold if we allow more general coefficient modules $\operatorname{Sym}^k \mathbf{Z}_p^2$ (regarded as a representation of $\Gamma_1(N)$, and thus as a sheaf on $Y_1(N)$, in the usual way). I know several references that treat this case, but they all assume $\ell \ne p$, while the original Ihara lemma still works for $\ell = p$.

Is the lemma still true in this more general setting?