In the theory of mod-$p$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in greater generality, and later on a different proof for $\ell \geq 5$ was accomplished in the work of Ash and Stevens: namely, the fact that for a rational prime $\ell$, the systems of Hecke eigenvalues of mod-$\ell$ modular forms of (say) level 1 coming from modular forms of weight $k \geq 2$ can all be produced (up to some theta twist) as mod-$\ell$ reductions of systems of Hecke eigenvalues of modular forms of weight $2 \leq k \leq \ell + 1$.

Is there a generalization of this fact to Hilbert modular forms ? I imagine that it will be a little bit more technically complicated: for example we would need to probably replace the condition $k \geq 2$ with the condition that the mod-$\ell$ Hilbert modular form in question is liftable to characteristic zero (as far as I am concerned this is fine). By Googling, I found the long paper of Andreatta and Goren, in which a similar thing is proved for forms that are ordinary (Hilbert modular forms: mod $p$ and $p$-adic aspects, Corollary 18.15). Is this result false as stated without the ordinary hypothesis ? Is there a salvage ?

Is there a fundamental reason why we cannot use the approach of Ash–Stevens to study this problem ?