Discrete series representations of quaternionic ${\mathrm{G}\mathrm{L}}_n(D)$ with symplectic periods

For a non-Archimedean locally compact field $F$ of odd residue characteristic and characteristic $0$, we prove a conjecture of D. Prasad predicting that, for an integer $n\ge 1$ and a non-split quaternionic $F$-algebra $D$, a discrete series representation of ${\mathrm{G}\mathrm{L}}_n(D)$ has a symplectic period if and only if it is cuspidal and its Jacquet–Langlands transfer to ${\mathrm{G}\mathrm{L}}_{2n}(F)$ is non-cuspidal.