Conjecture on knot Floer homology of Piccirillo friends

Let $K$ be a homologically thin slice knot with genus $g\ge 2$ and unknotting number $1$. Assume that $K$ has a positive unknotting crossing, and let $J$ be the corresponding Piccirillo friend. The corresponding exotic $\mathbb{R}^4$ pairs are formed from these knots.

Knot Floer homology conjecture. The group $\widehat{HFK}(J)$ is supported nontrivially in exactly $g$ diagonals, shifted up in Maslov grading from $K$. Hence the corresponding exotic $\mathbb{R}^4$ pairs are distinct.

This conjecture predicts a precise relationship between the knot Floer homology of a homologically thin slice knot and that of its Piccirillo friend, and would distinguish the associated exotic $\mathbb{R}^4$'s.