Finiteness conjecture for singular S-units

A singular modulus is a complex number that is the $j$-invariant of an elliptic curve with complex multiplication. For a finite set $S$ of rational primes, an algebraic number is an $S$-unit if all prime divisors of its principal ideal lie above primes in $S$. For a fixed finite set $S$ of rational primes, let the singular $S$-units be the singular moduli that are $S$-units.

Finiteness conjecture for singular S-units. For every $s \in \mathbb{N}$, the number of singular moduli that are $S$-units for some set of rational primes $S$ with $\#S=s$ is finite.

This conjecture predicts finiteness uniformly as the set of allowed primes varies, with the number of allowed primes fixed. The paper presents it after noting finiteness results for particular fixed sets of primes, but gives no resolution of the general statement.