Skewing identities for hook-indexed modules

For each positive integer $n$, let $\mathcal{S}_\rho$ denote the graded Frobenius characteristic associated with the hook shape $\rho$, let $e_1^\perp$ be the operator adjoint to multiplication by $e_1$, and let $\operatorname{Id}$ be the identity operator. Skewing conjecture. For all $n$,

$$(\operatorname{Id}\otimes e_1^\perp)\mathcal{S}_{(n)}=\sum_{a=0}^{n-2}\mathcal{S}_{(a\,|\,n-a-2)},$$

and

$$(e_1^\perp\otimes\operatorname{Id})\mathcal{S}_{1^n}=\sum_{a=1}^{n-1}\mathcal{S}_{(a\,|\,n-a-1)}.$$

These identities are reported as confirmed by computations for $n\leq4$, but no proof or general resolution is supplied.