Cuspidal representations of quaternionic $GL_n(D)$ with symplectic periods

Abstract: We prove a conjecture of Prasad predicting that a cuspidal representation of $GL_n(D)$, for an integer $n > 1$ and a non-split quaternion algebra $D$ over a non-Archimedean locally compact field $F$ of odd residue characteristic, has a symplectic period if and only if its Jacquet--Langlands transfer to $GL_{2n}(F)$ is non-cuspidal.