Automorphic tensor decomposition conjecture for non-induced cuspidal representations

Let $F$ be a number field and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbf{A}_F)$ that is not automorphically induced from any finite extension $L/F$. Automorphic tensor decomposition conjecture. There exist cuspidal automorphic representations $\tau$ and $\omega$ of $\mathrm{GL}_d(\mathbf{A}_F)$ and $\mathrm{GL}_{n/d}(\mathbf{A}_F)$, respectively, such that

$$\pi=\tau\boxtimes\omega,$$

with $\mathrm{BC}_{L/F}(\tau)$ cuspidal for every finite extension $L/F$, while for some finite extension $L/F$, $\mathrm{BC}_{L/F}(\omega)$ is the isobaric sum of $n/d$ copies of the trivial representation. The conjecture is motivated by the analogous structure theorem for Weil-group representations and requires deep functoriality; the source does not resolve it.