Real Floer realization conjecture for connected sums and doubly periodic knots

Let $K$ be any knot in $S^3$, let $K^r$ denote its reverse, and let $\fra$ denote auxiliary data. Let $\fro$ denote the auxiliary data associated with the doubly periodic knot $K\#K$. The notation $A$ is the ordinary Alexander grading and $A^R$ is the real Alexander grading.

Real Floer realization conjecture. There is an isomorphism of bigraded groups

$$\widehat{\operatorname{HFK}}(K)\cong\widehat{\operatorname{HFKR}}(K\#\tau K^r,\fra),$$

under the correspondence $A^R=\frac{1}{2}A$, for any choice of auxiliary data. Moreover,

$$\operatorname{rank}\widehat{\operatorname{HFK}}(K)=\operatorname{rank}\widehat{\operatorname{HFKR}}(K\#K,\fro),$$

for either choice of orientation.

The claim is motivated by computations for $K=3_1,4_1,5_1,$ and $5_2$. The supplied text reports these examples but gives no general proof or resolution.