The ring-structure reconstruction conjecture for the prismatized sphere

Let $\bbS^\mathrm{Syn}$ be the syntomic stack and let $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn}$ be its associated monoid stack. Let $(\bbS^\mathrm{Syn})_{\Spf \bbZ_p}$ and $(\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn})_{\Spf \bbZ_p}$ denote their pullbacks to $\Spf \bbZ_p$. Ring-structure reconstruction conjecture. The map

$$\bbZ_p^\mathrm{Syn}\to(\bbS^\mathrm{Syn})_{\Spf\bbZ_p}$$

identifies the source with the stack of ring structures on $(\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn})_{\Spf\bbZ_p}$. This is expected to follow from the main theorem cited in the source together with the non-functoriality of Raksit's height at least one deformed filtered de Rham complexes for rings.