A counterexample to a conjecture of complete fan

If a Griffiths domain D is a symmetric Hermitian domain, the toroidal compactification of the quotient space Gamma\D, associated to a projective fan and a discrete subgroup Gamma of Aut(D), was constructed by Mumford et al. Kazuya Kato and Sampei Usui studied extensions of Gamma\D for a Griffiths domain D in general, and introduced a notion of "complete fan" as a generalization of a notion of projective fan. The existence of complete fans is expected. In this paper, we give an example of D which has no complete fan.