On Zhang-Yang's open question about iterative roots of PM functions of height 1 (I): Increasing case

An open question was raised by Zhang-Yang in 1983: Does a PM function F with height 1 have an iterative root f of an order n≤N(F)+1 if the ‘characteristic endpoints condition’ is not satisfied? When F is strictly increasing on its characteristic interval K(F), this question was answered in the case n∈{N(F),N(F)+1} and the case n≤N(F)+1 for f increasing on K(F) and for f decreasing on K(F) with H(f)