On a minimal factorization conjecture

Let ϕ : S → D 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 , ϕ : S → D admits a deformation ϕ ′ : S ′ → D whose singular fibers are all of simple Lefschetz type. It has been conjectured that the factorization of the monodromy f ∈ M g around ϕ −1 ( 0 ) in terms of right-handed Dehn twists induced by the monodromy of ϕ ′ : S ′ → D 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 .