The symmetric-function identity for the fall version

For $n,k\in\mathbb{N}$, let $\Theta_{e_k}$, $\nabla$, and $\omega$ be the indicated symmetric-function operators, let $p_n$ and $p_{n,n(k+1)}$ be the relevant symmetric functions, and let $s_{(n-1)^k}^{\perp}$ denote the skewing operator. Fall-version symmetric-function identity.

$$\Theta_{e_k}\nabla\omega(p_n)=s_{(n-1)^k}^{\perp}p_{n,n(k+1)}.$$

The source states this identity after asserting that the right-hand side of the fall conjecture already has a formula, but supplies no proof or resolution; it remains open here.