Positivity for the $C_\alpha$ refinement

Let $\Lambda$ be the ring of symmetric functions, let $\nabla$ and the Delta operators act on it, and let $\phi$ be defined by

$$\phi f[X]=f[MX].$$

For a composition $\alpha=(\alpha_1,\ldots,\alpha_{\ell(\alpha)})$, define

$$C_\alpha=\mathbb{C}_{\alpha_1}\mathbb{C}_{\alpha_2}\cdots\mathbb{C}_{\alpha_{\ell(\alpha)}}(1),$$

where the operators $\mathbb{C}_m$ are those of Haglund, Morse, and Zabrocki. Positivity conjecture. For any composition $\alpha\vDash n-\ell$,

$$\left\langle \Delta_{\phi\Delta_{e_{\ell}}\phi^{-1}(h_pe_{n-d})}\nabla C_\alpha,h_{n-\ell}\right\rangle\in\mathbb{N}[q,t].$$

This conjecture predicts coefficientwise positivity in $q$ and $t$ for a compositional refinement of the generalized Delta setting. It is presented as computer-supported evidence, and no resolution is supplied in the source.