Hayden–Sabloff's quasipositivity criterion for exact Lagrangian fillability

A knot is quasipositive if it has the corresponding quasipositive braid or surface representation. Let $\operatorname{TB}(K)$ denote the maximal Thurston–Bennequin number of a knot $K$, and let $P_K(v,z)$ denote its HOMFLYPT polynomial. The HOMFLYPT bound is sharp when

$$\operatorname{TB}(K)=-\max\operatorname{deg}_{v}P_K(v,z)-1.$$

Hayden–Sabloff's conjecture. A knot is exact Lagrangian fillable if and only if it is quasipositive and the HOMFLYPT bound on the maximal Thurston–Bennequin number of $K$ is sharp.

The source notes that exact Lagrangian fillability implies quasipositivity and sharpness of the HOMFLYPT bound through results on transverse knots, augmentations, rulings, and Legendrian representatives. The converse is stated as a conjecture, with no resolution given here.