General Milnor conjecture for fibered links in tight contact 3-manifolds

Let $Y$ be a closed oriented 3-manifold, let $L\subset Y$ be a fibered link with fiber $\Sigma$, and let $\xi_L$ be the contact structure induced by its associated open book decomposition. Consider smoothly embedded surfaces in $Y\times[0,1]$ with boundary $L$ and carrying the relative homology class $[\Sigma]$.

General Milnor conjecture. If $\xi_L$ is tight, then $\Sigma$ maximizes Euler characteristic among all such surfaces.

This is the analogue for arbitrary tight contact structures of the fiber-surface genus-minimizing property for fibered links supporting the standard contact structure on the 3-sphere. The result is proved when the associated contact structure has non-vanishing Ozsváth–Szabó contact invariant, while the general tight case remains open.