Residual torsion-free nilpotence criterion for genus one pretzel knots

Let $J$ be a genus one pretzel knot, let $M_J$ denote its knot complement, and let $\Delta_J$ denote its Alexander polynomial. The commutator subgroup of $\pi_1(M_J)$ is residually torsion-free nilpotent if every nontrivial element survives in some torsion-free nilpotent quotient.

Residual torsion-free nilpotence conjecture. The commutator subgroup of $\pi_1(M_J)$ is residually torsion-free nilpotent if and only if $\Delta_J$ is nontrivial.

Proposition is the only known obstruction, and the exceptional genus one pretzel knots with nontrivial Alexander polynomial remain unresolved by the techniques of the paper. The conjecture predicts that the Alexander polynomial gives the precise criterion for residual torsion-free nilpotence in this setting.