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 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 $\operatorname{GL}_2$, which I am unable to understand because of the language.