The differentiable nonabelian Hodge correspondence for relative moduli spaces

Let $S$ be a smooth complex curve and let $X\to S$ be a polarized smooth family of smooth projective curves of genus at least $2$. Let $f:M^{s}_{dR}(X/S,r)\to S$ parametrize rank-$r$ irreducible holomorphic connections on the fibers, and let $g:M^{s}_{Hig}(X/S,r)\to S$ parametrize rank-$r$ stable Higgs bundles of degree zero. The nonabelian Hodge correspondence gives a canonical homeomorphism

$$\alpha_S:M^{s}_{dR}(X/S,r)\cong M^{s}_{Hig}(X/S,r).$$

Differentiable nonabelian Hodge conjecture. The map $\alpha_S$ is an isomorphism of differentiable fiber bundles over $S$. This would upgrade the known fiberwise topological identification to a differentiable one, but the supplied text gives no evidence that the conjecture has been resolved.