Minimum-genus Bennequin surface conjecture for transverse links

Let $\mathcal{T}$ be a transverse link in $(S^3,\xi_{std})$, and let $\delta(\mathcal{T})$ denote half the defect of the Bennequin inequality. A braid word representative $w$ has a Bennequin surface $F_w$, whose negative bands are the negatively twisted bands in its disk-band decomposition.

Minimum-genus Bennequin surface conjecture. Every transverse link $\mathcal{T}$ in $(S^3,\xi_{std})$ is represented by a braid word $w$ whose Bennequin surface $F_w$ contains $\delta(\mathcal{T})$ negative bands. Equivalently, every $\mathcal{T}$ bounds a Bennequin surface of genus $g(\mathcal{T})$.

This conjecture asks whether the defect exactly measures the minimum number of negative bands needed in a Bennequin surface. The supplied text notes a result under a condition of large fractional Dehn twist coefficient, but does not state a general resolution.