Torus-knot conjecture for nontrivial SU(2)-averse knots

Let $K$ be a nontrivial $SU(2)$-averse knot, meaning a knot with the $SU(2)$-averseness property discussed above.

Torus-knot conjecture. If $K$ is a nontrivial $SU(2)$-averse knot, then $K$ is a torus knot.

This conjecture predicts that torus knots are the only nontrivial knots with the $SU(2)$-averse property. The preceding results describe restrictions on $SU(2)$-cyclic surgery slopes and satellite constructions, but do not settle whether other $SU(2)$-averse knots exist.