The depth inequality and equality conjecture for Langlands packets

Let $G$ be a connected reductive group over $F$ that splits over $F^t$, let $\varphi$ be a positive-depth Langlands parameter for $G(F)$, and let $\Pi(\varphi)$ be its associated $L$-packet. Write $\rho(\varphi)$ for the depth of $\varphi$ and $\rho(\pi)$ for the depth of a representation $\pi$. Depth conjecture. For every $\pi\in\Pi(\varphi)$,

$$\rho(\varphi)\leq\rho(\pi).$$

If $p$ does not divide the order of the absolute Weyl group of $G$, then

$$\rho(\varphi)=\rho(\pi).$$

The conjecture is motivated by known equalities and inequalities for several classes of groups, together with examples where the inequality is strict; the source gives no general proof or disproof.