Finiteness conjecture for quasi-geometric connections with fixed Riemann symbol

Let $S$ be a finite subset of $\boldsymbol{P}^1$ and let $P$ be a quasiunipotent Riemann symbol for $S$, meaning the list of eigenvalues of monodromies around points of $S$. A connection on $\boldsymbol{P}^1\setminus S$ is quasi-geometric when it has regular singularities, is defined over $\overline{\Bbb Q}$, and its Betti and de Rham realizations are defined over $\overline{\Bbb Q}$. Finiteness conjecture. For fixed $P$, there are finitely many quasi-geometric connections on $\boldsymbol{P}^1\setminus S$ with Riemann symbol $P$. This is motivated by Deligne's corresponding finiteness result for geometric connections; the conjecture is presented as open, and would imply that most rank-two connections with $|S|=4$ and fixed $P$ are not quasi-geometric.