Rationality conjecture for $[200~\mathbb{Q}$-factorial nodal quartic double solids

A nodal quartic double solid is a double cover of $\mathbb{P}^3$ branched over a nodal quartic surface; it is $\mathbb{Q}$-factorial if every Weil divisor has a positive multiple that is Cartier.

Rationality conjecture for $\mathbb{Q}$-factorial nodal quartic double solids. Every $\mathbb{Q}$-factorial nodal quartic double solid is irrational.

The preceding results show that non-$\mathbb{Q}$-factorial nodal quartic double solids are rational except for the six-nodal example arising from a cubic threefold. The claim concerns the remaining $\mathbb{Q}$-factorial case and is presented as a generalization of the paper's rationality results.