Calabi–Yau theorem and complex Monge–Ampère equation for transversally Kähler manifolds

Question

Let $M$ be a compact smooth manifold, and $F\subset TM$ a smooth foliation. It is called transversally Kähler if the normal bundle $TM/F$ is equipped with a Hermitian structure (that is, a complex structure and a Hermitian metric) which is locally obtained as a pullback of a Kähler structure on the leaf space. Sasakian manifolds are prime examples of transversally Kähler manifolds (the leaf space of the Reeb foliation on a Sasakian manifolds is Kähler).

A differential form on $M$ is called basic if it vanishes on $F$ and is locally obtained as a pullback of a form on the leaf space. The basic forms are preserved by de Rham differential, and the cohomology of the basic forms is called the basic cohomology.

A foliation is taut if the top basic cohomology is non-zero (this is not the usual definition, but a theorem of Habib and Richardson). For taut foliations, one has also Poincaré duality on the basic cohomology, and the identification between the basic cohomology and the basic harmonic forms if a transversal Riemannian structure is given.

When $F$ is taut and transversally Kähler, the basic cohomology satisfy all the nice properties of the cohomology of the Kähler manifolds: the dd^c-lemma, the Hodge decomposition, the Hodge structure, Lefschetz SL(2)-action and so on. This is a folklore result of sorts, which follows (without much difficulty) from the Habib–Richardson theorem identifying the basic cohomology and basic harmonic forms.

I think I can prove the Calabi–Yau theorem (that is, the uniqueness and existence of the solutuions of the complex Monge–Ampère equation) for the basic forms; the proof is essentially the same as the usual Calabi–Yau, with the only (minor) modification in the $C^0$-estimates. The result is as follows.

THEOREM: Let $V$ be a basic volume form, and $\omega$ a transversal Kähler form on a taut foliated manifold $(M, F)$, with $\operatorname{dim}(M) -\operatorname{rk}(F)=k$. Assume that $\int_{M/F} V = \int_{M/F} \omega^k$, where $\int_{M/F}$ denotes the projection to the top degree basic cohomology class, given by the tautness of $F$. Then there exists a basic function $\phi$ such that $(\omega+dd^c \phi)^k=V$, and $\phi$ is unique up to a constant.

I suspect the result is already known in some other disguise. Could someone please enlighten me if this is known, and what are the good terms I should look in Google for this kind of stuff? From the search it is painfully obvious that "transversally Kähler" is something that only Sasakian people know and use. I suspect some other term is commonly used instead of "transversally Kähler", because the notion is ubiquitous.