Is [Scholl, *Motives for modular forms*, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?

Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing the level of $f$. Then the $\pi$-adic ($\pi \mid p$) realization is crystalline at $p$ and the characteristic polynomial equal to the Hecke polynomial of $f$ for $p$.

[Scholl] says that forthcoming work of Faltings should remove his condition $p \geq w$, but I read that there might be a mistake in Faltings' article. Is this true?

If it is true and known (which is what several posts like Galois representations attached to newforms here suggest), what is a citable reference (article and theorem therein)?