Przytycki's torsion conjecture for Kauffman bracket skein modules

Let $M$ be a compact oriented $3$-manifold. The Kauffman bracket skein module $\mathcal{S}(M)$ is the quotient of the free $\mathbb{Z}[A^{\pm1}]$-module generated by isotopy classes of framed links in $M$ by the Kauffman bracket relations.

Przytycki's torsion conjecture. The following are equivalent:

1. Every two-sided, closed, essential surface in $M$ is parallel to the boundary.
2. $\mathcal{S}(M)$ is torsion-free.

This conjecture relates torsion in the skein module to the presence of incompressible surfaces. Przytycki conjectured the implication from (2) to (1), while the reverse implication was later conjectured by Kalfagianni, Sikora, and the second author. The paper is concerned with proving the implication from (2) to (1), so the full equivalence remains open in the supplied text.