On the existence of admissible supersingular representations of $p$-adic reductive groups

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F\mathrm{/}{\mathbb{Q}}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.