Linear periods for unitary representations

Let $F$ be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadi'{c}'s classification of the unitary dual of ${\mathrm{G}\mathrm{L}}_{2n}(F)$, we classify irreducible unitary representations of ${\mathrm{G}\mathrm{L}}_{2n}(F)$ that have nonzero linear periods, in terms of Speh representations that have nonzero periods. We also give a necessary and sufficient condition for the existence of a nonzero linear period for a Speh representation.