The two-phase problem for harmonic measure in VMO and the chord-arc condition

Let Ω^+⊂ℝ^n+1 be a bounded δ-Reifenberg flat domain, with δ>0 small enough, possibly with locally infinite surface measure. Assume also that Ω^-= ℝ^n+1∖Ω^+ is an NTA domain as well and denote by ω^+ and ω^- the respective harmonic measures of Ω^+ and Ω^- with poles p^±∈Ω^±. In this paper we show that the condition that logdω^-dω^+∈ VMO(ω^+) is equivalent to Ω^+ being a chord-arc domain with inner normal belonging to VMO(H^n|_∂Ω^+).