Local finiteness of Grothendieck monoids of profinitary exact categories

Let $\mathscr A$ be a profinitary exact category, and let $\Gamma_{\mathscr A}$ be its Grothendieck monoid. A monoid is locally finite if, for every $\gamma\in\Gamma_{\mathscr A}$, there are only finitely many pairs $(\alpha,\beta)\in\Gamma_{\mathscr A}\times\Gamma_{\mathscr A}$ such that $\alpha+\beta=\gamma$. Local finiteness conjecture. The Grothendieck monoid $\Gamma_{\mathscr A}$ is locally finite. This would imply, by the preceding characterization, that every profinitary exact category is cofinitary; the claim is proposed beyond the classes already established in the paper.