Finiteness conjecture for thin hyperbolic hypergeometric monodromy groups

Let $H(\alpha,\beta)$ denote a hyperbolic hypergeometric monodromy group, where $\alpha$ and $\beta$ are the parameter tuples defining the corresponding hypergeometric equation. A subgroup of an arithmetic group is thin if it has infinite index in its arithmetic Zariski closure. Thinness conjecture. All but finitely many hyperbolic hypergeometric groups $H(\alpha,\beta)$ are thin. The paper establishes thinness for two infinite families with odd $n\geq 5$ and for various additional families and sporadic examples, but leaves open whether only finitely many hyperbolic hypergeometric monodromy groups fail to be thin. The accompanying remark notes that it is possible that all such groups are thin for $n\geq 5$.