Genericity transfer conjecture for unitary-group representations

Let $G/\mathbb{Q}$ be a unitary group becoming an inner form of $GL_n$ over an imaginary quadratic field $E$, with $G(\mathbb{R})$ compact. Let $p_1,p_2\ne l$ be distinct primes such that $G(\mathbb{Q}_{p_i})\cong GL_n(\mathbb{Q}_{p_i})$, and let $U\subset G(\mathbb{A}^{p_1,p_2})$ be open compact. Consider the representation on locally constant $\overline{\mathbb{F}}_l$-valued functions on the indicated double quotient.

Genericity transfer conjecture. If $\pi_1\otimes\pi_2$ is an irreducible subrepresentation of this function space and $\pi_1$ is generic, then $\pi_2$ is also generic.

This is proposed as an analogue of Ihara-type lemmas needed in the theory of automorphy lifting. The source explicitly presents it as a question formulated for potential use; no resolution status is given.