The conjecture that satellite knots are not quasi-alternating

Let a satellite knot be a knot whose exterior contains an incompressible torus. Let $\mathcal{Q}$ denote the class of quasi-alternating links, defined as the smallest set containing the unknot and closed under the crossing-smoothing recursion with determinant additivity. Satellite knots are not quasi-alternating.

This conjecture extends the known fact that prime alternating knots are not satellite knots in the context of quasi-alternating knots. Gordon and Lidman proved that cable knots have double branched covers that are not $L$-spaces, so cable knots are not quasi-alternating; the conjecture remains open for satellite knots in general.