Linear periods and distinguished local parameters

Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic. Let $X$ be the $p$-adic symmetric space $X = H \backslash G$,where$G = $\mathbf{G}\mathbf{L}$_{2n}(F)$and$H = $\mathbf{G}\mathbf{L}$_n(F) \times $\mathbf{G}\mathbf{L}$_n(F)$. We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of $X$.