The genus-one open-book fillability conjecture

Let $(M,\xi)$ be a closed contact three-manifold supported by a genus one open book $(S_{1,r},\phi)$, where $\phi$ is pseudo-Anosov and the associated foliations have two singularities on every boundary component of $S_{1,r}$. Suppose that $b_1(M)=0$. Genus-one open-book fillability conjecture. The contact structure $\xi$ is strongly symplectically fillable by a filling $W$ with $b_2^+(W)>0$ if and only if there exists some $d \in \mathbb{N}$ for which $c^+(\xi)$ is not in the image of $U^d$. The conjecture combines the known obstruction from the Ozsváth–Szabó contact invariant with the preceding corollary, conditional on the question of weak symplectic fillability for the relevant open books; its status is not resolved here.