Szabó's Conway mutation conjecture for knot Floer homology

Let $K$ and $K'$ be knots related by Conway mutation, and let the $\delta$-grading on $\widehat{HFK}$ be the difference between the Alexander and homological gradings. Szabó's conjecture. The rank of knot Floer homology is unchanged by Conway mutation in each $\delta$-grading:

$$\operatorname{rk}\,\widehat{HFK}_{\delta}(K)=\operatorname{rk}\,\widehat{HFK}_{\delta}(K')$$

for every $\delta$. The conjecture was communicated by Zoltán Szabó and is supported by computational evidence; the source gives no resolution status.