Open-book subsurface Thurston norm conjecture for tight contact manifolds

Let $(Y,\xi)$ be a closed contact 3-manifold. A supporting open book for $(Y,\xi)$ has a page, and a subsurface of that page has a boundary in $Y$. The Thurston norm measures the minimal complexity of a relative homology class represented by a surface.

Open-book subsurface conjecture. $(Y,\xi)$ is tight if and only if every subsurface of the page of each supporting open book is Thurston norm minimizing and, moreover, has minimal complexity for its boundary among all surfaces in $Y\times[0,1]$.

This conjecture characterizes tightness through four-dimensional complexity-minimizing properties of subsurfaces of supporting open books. The paper proves the corresponding minimization statement for subsurfaces when the supported contact structure has non-vanishing Ozsváth–Szabó contact invariant; the asserted equivalence for all tight contact structures remains open.