The minimum-weight-sector conjecture for veering triangulations

Let $M$ be a mapping torus equipped with a veering triangulation. A minimum weight sector is a sector of minimum weight in the associated veering data, and the hypothesis of

is the condition required there. **Minimum-weight-sector conjecture.** There \exists a fiber surface of $M$ such that the hypothesis of

is satisfied for some minimum weight sector. The conjecture would make the stronger double-hook bound universally applicable, potentially improving bounds for veering triangulations and normalized dilatations. It remains open in the supplied text.