The classical-family transfer conjecture for refined RASCARs on $\mathrm{GL}_4$

Let $\tilde\pi$ be a $p$-refined RASCAR of $\mathrm{GL}_4$, and let $P_{\tilde\pi}$ be its associated parabolic. A classical family is a family through $\tilde\pi$ consisting of classical points. Classical-family transfer conjecture. Every classical family through $\tilde\pi$ is the transfer of a classical family on $\mathrm{GSp}_4$, varies over a $P_{\tilde\pi}$-parabolic weight space, and has dimension $\#X_{P_{\tilde\pi}}+1$. This is presented as a conjectural strengthening in the $\mathrm{GL}_4$ case; the paper explains that the broader assertion that every classical family is symplectic is not stated as a formal conjecture.