Conjecture on transfers of test functions

Let $E \geq F' \geq F$ be a tower of number fields, let $S$ be a finite set of places of $F$ containing the infinite places, and let $S'$ and $S_0$ be the places of $F'$ and $E$ above $S$. Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$, and suppose $\Pi^{\sigma}\cong\Pi$ for every $\sigma\in\mathrm{Gal}(E/F)$. A transfer is a triple of compactly supported smooth functions $f_{S_0}$, $h_{S'}$, and $\Phi_S$ satisfying the trace identity of Definition of the paper for every irreducible generic unitary representation $\pi_S$ of $\mathrm{GL}_n(F_S)$; a function is of positive type in the sense used there. Transfer conjecture. There exist $f_{S_0}\in C_c^{\infty}(\mathrm{GL}_n(E_{S_0}))$ and $h_{S'}\in C_c^{\infty}(\mathrm{GL}_n(F'_{S'}))$ admitting a transfer $\Phi_S\in C_c^{\infty}(\mathrm{GL}_n(F_S))$ of positive type such that the transfer identity holds for every irreducible generic unitary representation $\pi_S$ of $\mathrm{GL}_n(F_S)$ and

$$\mathrm{tr}(\Pi_{S_0})(f_{S_0})\neq 0.$$

The conjecture asserts that sufficiently many transfers can detect every Galois-invariant cuspidal representation; the paper notes that it is known in some special settings, including $n=2$ and cyclic extensions of prime degree, but remains open in general.