Split embedding problems over complete domains

We prove that every finite split embedding problem is solvable over the field K((X₁, ..., X_n)) of formal power series in n ≥ 2 variables over an arbitrary field K, as well as over the field Quot.A[[X₁, ..., X_n]] of formal power series in n ≥ 1 variables over a Noetherian integrally closed domain A. This generalizes a theorem of Harbater and Stevenson, who settled the case K((X₁, X₂)).