Refined $cTC^-$ conjecture for the prismatization of the sphere

Let $\bbS^{\widehat{\calN}}$ be the stack of one-dimensional commutative formal groups with a section, and let $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^{\widehat{\calN}}$ be its associated monoid stack. Regard both as analytic stacks. Let $\TC^{-,\mathrm{ref}}$ denote Efimov's refined $\TC^-$. Refined $\TC^-$ conjecture. The graded cohomology ring of the analytic stack of $\bbQ$-algebra structures on $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^{\widehat{\calN}}$ is

$$\lim_\Delta \pi_{2*}\prn*{\TC^{-,\mathrm{ref}}(\bbQ) \tensor \MU^{\tensor (\bullet+1)}}.$$

The conjecture may require a slight modification of the monoid stack to make it analytically sensible, and it is unclear whether a strict ring structure must be imposed or follows automatically from $\bbQ$-linearity.