Let $G$ be a reductive group over a local non-archimedean field $F$.


Can every irreducible supercuspidal representation of $G(F)$ be realized as the induction from an open subgroup, which is compact modulo the center?



Are all irreducible supercuspidal representation induced from compact-mod-center subgroups?
It is known for GL(N) and SL(N) (Bushnell and Kutzko), for classical groups when the residue characteristic is not $2$ and when no quaternionic algebra is involved (Stevens), for GL(N) of a division algebra (Stevens and Sécherre), for a general reductive group when the residue characteristic of $F$ is large enough (Kim, Yu, ...). 


