Delta-prime formula for the bosonic coinvariant characteristic

Let $\mathcal{E}_n=\sum_{\mu\vdash n}\mathcal{C}_\mu\otimes s_\mu(\bm{z})$ be the pure bosonic multivariate coinvariant characteristic, where $\mathcal{C}_\mu=\sum_\lambda c_{\lambda\mu}s_\lambda$, and let $e_k^\perp$ denote the adjoint of multiplication by the elementary symmetric function $e_k$. Let $\Delta'_{e_{n-k-1}}$ be the Delta-prime operator. Delta via skew conjecture. For all $k$,

$$(e_k\otimes\operatorname{Id})\mathcal{E}_n=\sum_{\mu\vdash n}(e_k^\perp\mathcal{C}_\mu)(q,t)s_\mu(\bm{z})=\Delta'_{e_{n-k-1}}(e_n(\bm{z})).$$

The source identifies this as the precise link between the generic bosonic expression and the Delta-prime operators, referring to an earlier conjecture; no resolution is supplied in the given text.