In Sullivan's postscript to his MIT notes https://www.maths.ed.ac.uk/~v1ranick/surgery/gtop.pdf he describes some problems and conjectures, where Problem 4 is: "Analyze the action of Gal($\overline{\mathbb Q}/\mathbb Q$)on the smooth manifold structure set on a profinite homotopy type associated to nonsingular algebraic varieties defined over $\overline{\mathbb Q}$." This manifold structure set is on a profinite etale homotopy type obtained from nonsingular algebraic varieties over $\mathbb C$, though I confess I don't quite know what this means.

Nonetheless, in view of this he poses the Conjecture 4: "There is a concrete context with symmetry in structure which synthesizes these two compatible contexts, nonsingular simply connected algebraic varieties over $\overline{\mathbb Q}$ with $dim_{\mathbb C} > 2$ and Gal$\overline{\mathbb Q}/\mathbb Q$) symmetry, and simply connected topological manifolds with $dim_{\mathbb R}>4$ and the $\hat{\mathbb Z}^*$ symmetry on the invariants (defined using the isomorphic part of the Adams operations at odd primes from Part I and the cohomological construction of the ICM ‘70 paper andfootnote 8 there at the prime 2)"

I am wondering if there has been any progress (directly or indirectly related) to this conjecture since it was posed in 2004?