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It is conjectured that the Dolbeault cohomology of a complex nilmanifold   $X$   is computed by left-invariant forms. We prove this under the assumption that   $X$   is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalizes previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups