Full-faithfulness conjecture for the syntomic sphere and its monoid stack

Let $\bbS^\mathrm{Syn}$ be the syntomic stack, and let $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn}$ be the associated monoid stack. Consider the map from $\bbS^\mathrm{Syn}$ to the stack of abelian monoid stacks classifying $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn}$. Full-faithfulness conjecture. This map is fully faithful. The maximal subgroup of the monoid stack is tractable, and the stated difficulty is extending group homomorphisms to monoid homomorphisms.