I am not an expert in foliations, but I'm interested in the following conjecture of Cantwell and Conlon (from "An interesting class of $C^1$ foliations"): "If $F$ is a transversely orientable foliation of codimension one on a compact manifold $M$ such that the leaves are all at finite depths and if all junctures have quasi-polynomial growth, then $(M, F)$ is homeomorphic to a $C^1$-foliated manifold."
What would it take to show that one can smooth out a given foliation? More precisely: is it equivalent to showing that one can topologically conjugate the holonomy pseudogroup to a pseudogroup of $C^1$ diffeomorphisms?