On a minimal factorization conjecture

Let be a proper holomorphic map from a connected complex surface S onto the open unit disk D⊂C, with 0∈D as its unique singular value, and having fiber genus g>0. Assume that in case g⩾2, admits a deformation whose singular fibers are all of simple Lefschetz type. It has been conjectured that the factorization of the monodromy f∈Mg around ϕ−1(0) in terms of right-handed Dehn twists induced by the monodromy of has the least number of factors among all possible factorizations of f as a product of right-handed Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585–594]). In this article, the validity of this conjecture is established for g=1.