Symplectic Hecke insertion conjecture for FPF-involution words

Let $v$ and $w$ be FPF-involution words, and let $\overset{\textsf{Sp}}\sim$ denote the symplectic Hecke equivalence relation. The symplectic Hecke insertion algorithm assigns to each word $u$ a tableau $P_{\textsf{Sp}}(u)$. Symplectic Hecke insertion conjecture. If

$$v\overset{\textsf{Sp}}\sim w,$$

then

$$P_{\textsf{Sp}}(v)=P_{\textsf{Sp}}(w).$$

This would establish that symplectic Hecke insertion tableaux are invariants of the relevant equivalence classes of FPF-involution words; the supplied text gives computational and structural motivation but no resolution.