Maximal quasi-etaleness of general fibres

Determine whether a general fibre of the holomorphic projective MRC fibration associated with a compact klt Kähler variety admitting an int-amplified endomorphism is itself maximally quasi-etale.

Background

The paper establishes, after an equivariant quasi-etale cover, a flat holomorphic projective MRC fibration from a compact klt Kähler variety admitting an int-amplified endomorphism onto a complex torus. In the smooth case, the fibration is smooth and its periodic fibres are of Fano type.

For singular klt varieties, the authors explicitly note that the theorem does not establish maximal quasi-etaleness for general fibres. This issue concerns whether the quasi-etale and fundamental-group properties of the total space descend to the fibres and is left unresolved.