On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the ${\mathrm{G}\mathrm{L}}_{2n}$ eigenvariety

Friedberg–Jacquet proved that if $\pi$ is a cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_{2n}(\mathbb{A})$, then $\pi$ is a functorial transfer from ${\mathrm{G}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H={\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_n$ is non-vanishing on $\pi$. We conjecture a $p$-refined analogue: that any $P$-parahoric $p$-refinement ${\tilde{\pi }}^P$ is a functorial transfer from ${\mathrm{G}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{2n+1}$ if and only if a $P$-twisted version of $Z_H$ is non-vanishing on the ${\tilde{\pi }}^P$-eigenspace in $\pi$. This twisted $Z_H$ appears in all constructions of $p$-adic $L$-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the ${\mathrm{G}\mathrm{L}}_{2n}$ eigenvariety, and – by proving upper bounds on the dimensions of such families – obtain various results towards the conjecture.