Subregular Fourier-coefficient conjecture for generic representations of ${\mathrm {U}}_{n,n}$

Let $F$ be a number field, let ${\mathrm {U}}_{n,n}$ be the $F$-quasisplit unitary group, and let $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm {U}}_{n,n}({\mathbb A})$. Assume that $\pi$ is generic, meaning that it has a nonzero Whittaker-Fourier coefficient. Let

$$\underline{p}_{\mathrm{subr}}=[(2n-1)1]$$

be the subregular partition. Subregular Fourier-coefficient conjecture. The representation $\pi$ has a nonzero Fourier coefficient associated to $\underline{p}_{\mathrm{subr}}$. This conjectural generalization of the proved ${\mathrm {U}}_{2,2}$ proposition would extend the corresponding central-value non-vanishing theorem to general quasisplit unitary groups ${\mathrm {U}}_{n,n}$.