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

Friedberg--Jacquet proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A})$, $\pi$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H = \mathrm{GL}_n \times \mathrm{GL}_n$ 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 $\mathrm{GSpin}_{2n+1}$ 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 prove various results towards the conjecture by connecting it to the study of classical symplectic families in the $\mathrm{GL}_{2n}$ eigenvariety. If $\pi$ is spherical at $p$, there are $(2n)!$ attached $p$-refinements to Iwahori level; we conjecture the dimensions of such families through each of these, prove the upper bound unconditionally, and (by constructing the families) prove the lower bound under a non-critical slope assumption. For example, for $\mathrm{GL}_4$, we conjecture that (modulo 1-dimensional trivial variation) 8 refinements vary in 2-dimensional symplectic families, 8 in 1-dimensional symplectic families, and prove 8 do not vary in any symplectic family.