Tensor product functoriality conjecture for RAECSDC representations

Let $r\geq 1$. Let $F$ be a CM field, and let $(\pi_1,\psi_1)$ and $(\pi_2,\psi_2)$ be RAECSDC automorphic representations of $\mathrm{GL}_2({\mathbf A}_F)$ and $\mathrm{GL}_r({\mathbf A}_F)$, respectively. Fix a prime $p$ and an isomorphism $\iota:\overline{{\mathbf Q}}_p\to{\mathbf C}$. Let $\rho=r_\iota(\pi_1)\otimes r_\iota(\pi_2)$. Tensor product functoriality conjecture. If $\rho$ is irreducible and Hodge--Tate regular, then there exists a RAECSDC automorphic representation $(\pi_3,\psi_3)$ of $\mathrm{GL}_{2r}({\mathbf A}_F)$ such that

$$r_\iota(\pi_3)\cong\rho.$$

This is the tensor product functoriality assertion attributed in the source to Clozel. It predicts automorphy of the tensor product under the stated irreducibility and regularity hypotheses; no resolution is given here.