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 ?⁻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.