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I happen upon this old thread now. Maybe it is still worth giving an update, and more of an answer. The latest account (as of the time of this writing) of the conjectural statement in question here appears as Conjecture 1.17 in Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology in H. Sati, U. Schreiber (eds.) ...

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