Equivalence of domains arising from duality of orbits on flag manifolds

S. Gindikin and the author defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open -orbit on an ``arbitrary' flag manifold generalizing the result of Barchini. This conjecture for closed was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.