What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$?

Proposition 3.4 in [Loeffler and Weinstein - On the computation of local components of a newform](https://arxiv.org/abs/1008.2796) refers to [Breuil and Mézard - Multiplicités modulaires et représentations de $\operatorname{GL}_2(\mathbb Z_p)$
et de $\operatorname{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ en $\ell = p$. Appendice par Guy Henniart. Sur l'unicité des types pour $\operatorname{GL}_2$](https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/multiplicite.pdf), which I am unable to understand because of the language.