Classification conjecture for genus-1 simplified broken Lefschetz fibrations

Let $M$ be a $4$-manifold admitting a genus-$1$ simplified broken Lefschetz fibration (SBLF) structure with non-empty round singular locus. Write $\sharp$ for connected sum, and let $L_n$ and $L_n^{\prime}$ denote the manifolds used in the classification of genus-$1$ SBLFs. Classification conjecture. If $M$ admits such a structure, then it is diffeomorphic to one of the following $4$-manifolds:

• $\sharp k\mathbb{CP}^2\sharp l\overline{\mathbb{CP}^2}$, where $l>0$ and $k\geq 0$;
• $\sharp k(S^2\times S^2)$, where $k\geq 0$;
• $S^1\times S^3\sharp S\sharp k\overline{\mathbb{CP}^2}$, where $k\geq 0$ and $S$ is either $S^2\times S^2$ or $S^2\widetilde{\times}S^2$;
• $L\sharp k\overline{\mathbb{CP}^2}$, where $k\geq 0$ and $L$ is either $L_n$ or $L_n^{\prime}$.

More strongly, the families of genus-$1$ SBLFs obtained in the cited section should contain all genus-$1$ SBLFs with non-empty round singular locus. The statement proposes a complete diffeomorphism classification, but the supplied text gives no resolution status.