The Maslov-grading conjecture for strongly quasipositive knots

Let $K$ be a strongly quasipositive knot with genus $g$, and let $\widehat{\mathit{HFK}}_m(K,g)$ denote the summand of knot Floer homology in Maslov grading $m$ and top Alexander grading $g$. Maslov-grading conjecture. The minimal Maslov grading in the top Alexander piece is zero:

$$\min\{m\mid\widehat{\mathit{HFK}}_m(K,g) \neq 0\}=0.$$

This would extend the observed relationship between the leading exponent of $F_K$ and knot Floer homology beyond the classes checked in the paper. Its general validity is open.