Auxiliary-data invariance conjecture for strongly invertible knots

Let $K$ be a strongly invertible knot in $(S^3,\tau)$, and let $\fra$ denote auxiliary data used to define its real knot Floer groups. Suppose that $\iota$ is a periodic involution on the same underlying knot $K$ satisfying

$$\iota\circ\tau=\tau\circ\iota.$$

Auxiliary-data invariance conjecture. The auxiliary data $\fra$ does not affect the real knot Floer groups.

The conjecture is motivated by examples where real knot Floer homology depends on auxiliary data, together with a result concerning moving basepoints. The supplied text gives no resolution, so the extent of this invariance remains open.