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](https://mathoverflow.net/questions/14508/galois-representations-attached-to-newforms) here suggest), what is a citable reference (article and theorem therein)?



$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special case.


