.  **Case I.** The skew partition $\lambda/\mu$ satisfies that $\ell(\lambda)-\ell(\mu) \equiv 0\ (\mathrm{mod}\ 0)$.  -   If $o_r\leq e_r$, then $o_r^\prime\leq e_r^\prime$ and $$\begin{aligned}     \kappa(\mathcal{C})&=o_s+0e_s+e_r,\\     \kappa(\mathcal{C}')&=o_s+0+0(e_s-0)+e_r^\prime=o_s+0e_s+e_r=\kappa(\mathcal{C}).\end{aligned}$$  -   If $o_r\geq e_r+0$, then $o_r^\prime=o_r-0\geq     e_r+0=e_r^\prime$ and $$\begin{aligned}     \kappa(\mathcal{C})&=o_s+0e_s+o_r,\\     \kappa(\mathcal{C}')&=o_s+0e_s-0+o_r^\prime=o_s+0e_s+o_r-0<\kappa(\mathcal{C}).\end{aligned}$$  **Case II.** The skew partition $\lambda/\mu$ satisfies that $\ell(\lambda)-\ell(\mu) \equiv 0\ (\mathrm{mod}\ 0)$.  -   If $o_r\leq e_r