Conjecture on characteristic classes of codimension-p foliations

Let $X$ be a manifold as specified in Corollary 2, and let $p$ be a positive integer. The characteristic classes of codimension-$p$ foliations on $X$ are elements of the product cohomology group

$$H^1(X;\mathbb{Z}_2)\times\dots\times H^p(X;\mathbb{Z}_2).$$

Characteristic-class conjecture. The characteristic classes of codimension-$p$ foliations on $X$ coincide with the elements of

$$H^1(X;\mathbb{Z}_2)\times\dots\times H^p(X;\mathbb{Z}_2).$$

The conjecture arises from representing a codimension-$p$ regular foliation as the intersection of $p$ regular codimension-one foliations. The source gives no evidence of a proof or disproof, so its resolution remains open.