Symplectic Schur-times-Schur conjecture for skew semistandard oscillating tableaux

For partitions $\lambda,\mu,\nu$ of length at most $m$, let $\chi_\lambda$ be the irreducible character of $Sp(2m)$ indexed by $\lambda$, and let $s_\mu=s_\mu(x_1^{\pm},\ldots,x_m^{\pm})$ be the character of the restriction to $Sp(2m)$ of the irreducible $GL(2m)$-representation indexed by $\mu$. For a skew semistandard oscillating tableau $T$, write $\operatorname{inside}(T)$ and $\operatorname{outside}(T)$ for its inner and outer shapes, $\operatorname{wt}(T)$ for its weight, $c(T)$ for its column bound, and $\varepsilon_i(T)$ for the crystal statistic. Symplectic Schur-times-Schur conjecture. The number of copies of $\chi_\nu$ in $\chi_\lambda s_\mu$ equals the number of skew semistandard oscillating tableaux $T$ satisfying

$$\operatorname{inside}(T)=\lambda',\qquad \operatorname{outside}(T)=\nu',\qquad \operatorname{wt}(T)=\mu',\qquad c(T)\leq m,$$

and

$$\varepsilon_i(T)=0\quad\text{for }i=1,2,\ldots,m-1.$$

This conjecture generalizes the dual Pieri rule. The cases where $\mu=(k)$ or $\mu=(1^k)$ are known, while the general formula is presented as conjectural.