Distinction of some induced representations

Let $K/F$ be a quadratic extension of $p$-adic fields, $sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $pi^{vee}$ the smooth contragredient of $pi$, and by $pi^{sigma}$ the representation $picirc sigma$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $pi^vee otimes pi^sigma$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields.