A Riemann-Roch theorem in Bott-Chern cohomology

If $M$ is a complex manifold, the Bott-Chern cohomology $H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)$ of $M$ is a refinement of de Rham cohomology, that takes into account the $(p,q)$ grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript $\mathrm{BC}$. Let $p:M\to S$ be a proper holomorphic submersion of complex manifolds. Let $F$ be a holomorphic vector bundle on $M$ and let $Rp_{*}F$ be its direct image. We will prove that if $Rp_{*}F$ is locally free, then $$\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}F\right)=p_{*}\left[ \mathrm{Td}_{\mathrm{BC}}\left(TM/S\right) \mathrm{ch}_{\mathrm{BC}}\left(F\right)\right].$$ If $p$ is projective, this result is a consequence of the theorem of Riemann-Roch-Grothendieck. If $M$ is Kaehler, it follows from a families version of classical Hodge theory. In the general case, none of these tools is available. We will show how a suitable hypoelliptic deformation of Hodge theory can be used to prove the above result.