The rigid del Pezzo surface unicuspidal curve–almost toric fibration conjecture

Let $M$ be a rigid del Pezzo surface. An index zero $(p,q)$-unicuspidal rational symplectic curve in $M$ is as in the source, and an almost toric fibration is a fibration $obreak automatically$ with base $Q$ as in Theorem~. Rigid del Pezzo surface conjecture. The following are equivalent: there \exists an index zero $(p,q)$-unicuspidal rational symplectic curve in $M$; and there \exists an almost toric fibration $obreak\pi:\mathbb{A}\rightarrow Q$ as in Theorem~, where $obreak\mathbb{A}$ is diffeomorphic to $M$ and $Q$ has consecutive edges pointing in the directions $(-mq^2,mpq-1),(0,-1),(1,0)$ for some $m\in\mathbb{Z}_{\geq 1}$. In particular, there \exists an index zero $(p,q)$-unicuspidal rational algebraic curve in $M$. This proposes an analogue of the established correspondence for $F_1$ and $\mathbb{CP}^2$ on the remaining rigid del Pezzo surfaces; the source does not provide evidence resolving the equivalence.