Symplectic Sagan–Worley recording-tableau conjecture for modified Knuth operators

Let $a$ be a primed word and let $\mathsf{spkn}_i(a)$ be the modified symplectic Knuth operator, with $\mathsf{spkn}_i(a)=\mathsf{okn}_i(a)$ in the cases not separately modified. Let $Q^\mathsf{Sp}_{\textsf{SW}}(a)$ denote the recording tableau under symplectic Sagan–Worley insertion, and let $\mathfrak{d}_i$ denote the corresponding dual-equivalence operation.

Symplectic Sagan–Worley recording-tableau conjecture. If $i>0$ and $a$ is any primed word, then

$$Q^\mathsf{Sp}_{\textsf{SW}}(\mathsf{spkn}_i(a))=\mathfrak{d}_i(Q^\mathsf{Sp}_{\textsf{SW}}(a)).$$

The analogous statement for the insertion tableaux is known from Worley's characterization of symplectic Knuth equivalence. The paper notes that it does not know a reference for this recording-tableau analogue, so its status remains open.