More detail on Jones's 30 manifold and analogous constructions can be found in my Extended powers of manifolds and the Adams spectral sequence , Cont. Math. 271, 41--51. The basic idea in it dates from 1979 or so (immediately upon seeing Jones's construction) but didn't make it into print for another 20 years. There are also comments about the homotopy theoretic approach to the problem in the talk I gave in Edinburgh in 2011 at the workshop on the Kervaire invariant.
Robert Bruner
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