Boundary continuity of Dirichlet finite harmonic measures on compact bordered Riemannian manifolds (Q1357697)

\textit{S. Segawa} [Proc. Am. Math. Soc. 103, 177-183 (1988; Zbl 0644.30026)] has conjectured the following generalization of the Beurling projection theorem for the harmonic measure: Let \(K\subset [-1,1]\) be the union of a finite number of intervals of total length \(2m\). Then \(\omega (0,K,D \backslash K) \geq \omega (0,K^*,D \backslash K^*)\) where \(K^*= [-1,-1+m] \cup [1-m,1]\). In [\textit{M. Essén} and \textit{K. Haliste}, Complex Var. Theory Appl. 12, 137-152 (1989; Zbl 0645.31006)] it was shown that this holds whenever \(K\) is symmetric with respect to the imaginary axis. The author proves the conjecture when \(K\) consists of two slits and \(D\backslash K\) is a simply connected domain.