Factoriality conjecture for sextic double solids with few mild singularities

Let $B\subset\mathbb{P}^3$ be a sextic surface with at worst $A_m$ singularities, where $m\leq 10$, and let $X$ be the double solid branched along $B$. The singular points of $X$ correspond to the singularities of the branch surface.

Factoriality conjecture for sextic double solids. If $X$ has at most four singular points, then $X$ is factorial.

The conjecture is suggested by examples showing that certain sextic double solids with terminal $cA_1$ points are factorial, while another example with an $A_{11}$ singularity fails to be $\mathbb{Q}$-factorial. It proposes factoriality under both a bound on the singularity type and a bound on the number of singular points.