Let $\Sigma_{g,r}$ be the surface of genus $g$ and $r$ boundary components. It is known that, from a positive factorization of a mapping class $\phi$ in the mapping class group $MCG(\Sigma_{g,r}, \partial \Sigma_{g,r})$, one can construct a Lefschetz fibration (with corners) over the disc $D^2$ so that there are as many critical points/values as Dehn twists take place in the factorization and such that the restriction of the Lefschetz fibration to $\partial D^2$ is a surface bundle with monodromy $\phi$. Conversely, from any such fibration one can recover a mapping class factorized by right-handed Dehn twists.

My question is:

Is it known for which mapping classes $\phi \in MCG(\Sigma_{g,r}, \partial \Sigma_{g,r})$ there exists $h \in \mathbb{N}$ and a Lefschetz fibration over the surface $\Sigma_{h,1}$ such that the monodromy of the restriction of the Lefschetz fibration to $\partial \Sigma_{h,1}$ is $\phi$?


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