The Gross–Prasad multiplicity comparison inequality

Let $\pi_V\in\Pi_{\mathrm{CW}}(\operatorname{SO}(V))$ and $\pi_W\in\Pi_{\mathrm{CW}}(\operatorname{SO}(W))$. Let $X^+$ be the space defining $\operatorname{GL}(X^+)$, and let $\sigma$ be a generic representation of $\operatorname{GL}(X^+)$. Multiplicity comparison conjecture. One has

$$m(\pi_V\boxtimes\pi_W)\leq m\bigl((\sigma\rtimes\pi_W)\boxtimes\pi_V\bigr).$$

The inequality is proposed as a building block for the reverse inequality in the multiplicity formula and hence for the reduction to tempered representations. The source presents it as expected and does not establish it in the supplied text.