The homotopy-lifting conjecture for secondary constructions

Let $(M,C)$ be a foliated manifold, let $(\overline{\Lambda},\overline{d})$ be the differential graded commutative algebra of differential forms on $M$ longitudinal along $C$, and let $\overline{\mathfrak{X}}$ be the module of vector fields transversal to $C$. Secondary constructions are the cohomological constructions obtained from the complexes $(\overline{\Lambda},\overline{d})$ and $(\overline{\Lambda}\otimes\overline{\mathfrak{X}},\overline{d})$, including secondary functions, vector fields, differential forms, and the analogues of the standard Cartan calculus.

Homotopy-lifting conjecture. All secondary constructions come from suitable homotopy structures at the level of characteristic cochains.

The claim proposes that the algebraic structures visible in secondary calculus are induced by corresponding structures up to homotopy before passing to characteristic cohomology. The statement is presented as the aim of the paper and no resolution is supplied here.