The transfer conjecture for symplectic families of refined RASCARs

Let $\tilde\pi$ be a $p$-refined RASCAR of $\mathrm{GL}_{2n}$, and let $P_{\tilde\pi}$ be the associated parabolic. A symplectic family is a family through $\tilde\pi$ in the eigenvariety whose classical points are symplectic. Transfer conjecture for symplectic families. Every symplectic family through $\tilde\pi$ is the transfer of a classical parabolic family for $\mathrm{GSpin}_{2n+1}$, varies over $\sW_{0,\lambda_\pi}^{P_{\tilde\pi}}$, and has dimension $\#X_{P_{\tilde\pi}}+1$. The source proves a matching lower-bound result under non-critical slope and regularity hypotheses, but the assertion about every symplectic family remains conjectural.