Iterations of Stone-Čech remainders

Consider a compact space X=L0∪L1 which is a disjoint union of two Lindelöf dense subsets L0, L1, and fix this partition. Then we repeat taking remainders of Stone-Čech compactifications ω1 many times to construct a compact space Ω(X) which is “minimal” w.r.t. the property that Ω(X) admits an irreducible map π onto X such that both π−1(L0) and π−1(L1) are C⁎-embedded in Ω(X). A typical example is the closed interval [0,1]=Q∪P with the partition into the rationals Q and the irrationals P, and we can show that the resultant compact space Ω([0,1]) is not extremally disconnected, hence different from the absolute of [0,1]. We respond to one of van Douwen's questions in [2] that the length of iteration needed for Ω([0,1]) is the first uncountable ordinal ω1.