Uniformity 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 each $s \in \mathbb{N}$, let $\mathcal{J}_s$ denote the set of singular moduli that are $S$-units for some set $S$ of rational primes with $\#S=s$.

Uniformity conjecture for singular $S$-units. For every $s \in \mathbb{N}$, the set $\mathcal{J}_s$ is finite.

Equivalently, for every finite set $S$ of rational primes, the set of singular $S$-units is finite, with its cardinality bounded only in terms of $\#S$, independently of the primes in $S$. The conjecture is motivated by the paper's computations and is not resolved there.