  Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal
of its ring of integers of norm $N$ and define, for every even $n$, the
$[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that
for a non CM Hilbert cuspidal Hecke eigenform for $F$, say $f\in
S_{\textbf{k}}(\Gamma_0(\frak{N}))$ with $k>2$ even, and a prime
$p>\max\{k+1,6\}$ totally split in $F$ such that $p\nmid N$ and such that the
residual mod $p$ representation $\overline{\rho}_f$ satisfies that
$\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overline{\rho}_f)$, there
exists a lift $\rho_g$ associated to a Hilbert modular cuspform for $F$, say
$g\in S_{\textbf{2}}(\frak{N}p^2,\epsilon)$ for some Nebentypus character
$\epsilon$ which is supercuspidal at each prime of $F$ over $p$. We also
observe that our techniques provide an alternative proof to the corresponding
statement for classical Hecke cuspforms already proved by Khare \cite{khare}
with classical techniques. Finally, we take the opportunity to include a
corrigenda for \cite{dieulefait} which follows from our main result, which
provides a congruence that puts the micro good dihedral prime in the level.
