Nontrivial real knot Floer homology conjecture for strongly invertible knots

Let $K$ be any knot in $S^3$. A strong inversion is an involution of the knot having the standard strong-inversion property. A real knot Floer group is the group associated to a knot together with a chosen strong inversion.

Nontriviality conjecture. If $K$ admits a strong inversion, then there is a strong inversion on $K$ with nontrivial real knot Floer group.

Computations for knots with at most seven crossings motivate the claim: although real knot Floer groups can be trivial for nontrivial knots, the real Alexander polynomial distinguishes the tested strong inversions except for $7_7$. Whether every strongly invertible knot has some inversion with nontrivial real knot Floer homology remains open.