General slice-Bennequin inequality for tight contact 3-manifolds

Let $(Y,\xi)$ be an oriented contact 3-manifold, and let $\Sigma$ be a smoothly embedded surface in $Y\times[0,1]$ whose boundary is Legendrian in $Y\times\{1\}$. Here $tb_\xi(\partial\Sigma)$ is the Thurston–Bennequin invariant and $rot_{[\Sigma]}(\partial\Sigma)$ is the rotation number relative to $\Sigma$.

General slice-Bennequin inequality. $(Y,\xi)$ is tight if and only if

$$tb_\xi(\partial \Sigma)+|rot_{[\Sigma]}(\partial \Sigma)|\leq-\chi(\Sigma)$$

for every such $\Sigma$.

This conjecture proposes that tightness is characterized by a four-dimensional slice-Bennequin bound for surfaces in the product cobordism $Y\times[0,1]$, extending the classical inequality in the 3-sphere. The stated theorem for contact manifolds with non-vanishing Ozsváth–Szabó contact invariant provides evidence, but the general case remains open.