Minimum-genus alpha-Bennequin surface conjecture

Let $(S,\phi)$ be an open book decomposition supporting a contact $3$-manifold $(M,\xi)$. Let $\mathcal{T}$ be a null-homologous transverse link in $(M,\xi)$) with Seifert surface class $\alpha\in H_2(M,\mathcal{T};\mathbb{Z})$. Let $\delta(\mathcal{T},\alpha)$ denote the defect of the Bennequin–Eliashberg inequality with respect to $\alpha$.

Minimum-genus alpha-Bennequin surface conjecture. If $\delta(\mathcal{T},\alpha)\geq 0$, then $\mathcal{T}$ bounds a minimum-genus $\alpha$-Bennequin surface with respect to $(S,\phi)$.

The claim is the converse of the necessary condition established in the surrounding discussion: bounding a minimum-genus $\alpha$-Bennequin surface implies nonnegative defect. The supplied text does not state a general proof; it later explains that this conjecture would characterize tightness through such surfaces.