  Suppose that $F$ is a non-Archimedean local field and $D$ is a central
division algebra over $F$. Let $n$ be a positive integer. We show a
classification modulo essentially square-integrable representations of standard
modules of $\mathrm{GL}_n(D)$ which have non-zero linear periods. By this
classification, the conjecture of Prasad and Takloo-Bighash is reduced to the
case of essentially square integrable representations.
