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)"
This concrete context, according to Sullivan, is "something like replace a manifold or variety by the inverse system or limit of its finite (branched) covers to obtain solenoidal manifolds with branching singularities. The symmetries might be combinatorially defined by rearrangement of the branched covers."
I am wondering if there has been any progress (directly or indirectly related) to this conjecture since it was posed in 2004?
