Supercuspidal representations of ${\rm GL}_n(F)$ distinguished by a Galois involution

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $\sigma$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $\pi$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $\pi^{\sigma}\simeq\pi^{\vee}$ (such a representation is said to be $\sigma$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times}$. When $\pi$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $\pi$ to be ${\rm GL}_n(F_0)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $\omega$ the non-trivial $R$-character of $F_0^{\times}$ trivial on $F/F_0$-norms, we prove that any $\sigma$-selfdual supercuspidal $R$-representation is either distinguished or $\omega$-distinguished, but not both. In the modular case, that is when $\ell>0$, we give examples of $\sigma$-selfdual cuspidal non-supercuspidal representations which are not distinguished nor $\omega$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan-Kable-Tandon.