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@TerryTao I thought about approximating $1_B$ by polynomials, but the approximation (in the Besicovitch seminorm) does not seem good enough, specially considering that the von Mangoldt function is unbounded. Another way to think about it: one can read $1_B$ of a (1-step) nilsystem but with a discontinuous (yet Riemann integrable) function $F$.
It should be mentioned that the connection you refer to is due to Furstenberg (ams.org/mathscinet-getitem?mr=498471). Later Furstenberg and Katznelson together used this connection to derive other combinatorial results, including a multidimensional extension of Szemeredi's theorem and a density version of the Hales-Jewett's theorem.
Yes, that's the easy argument I had in mind. I suspected this might be open (as many other corollaries of Chowla's conjecture)... Thanks for the references on the progress so far. I am intrigued by the result you mentioned about progressions with arbitrary sign patterns, I don't see how it follows from Mobius orthogonality plus the inverse Gowers norm conjecture. I am also wondering if it comes with statistics about the "proportion" of progressions with a given sign pattern.
