Comparing the Edelman-Greene bijection to David Little's bijection

Question one now has an answer. Benjamin Young and I have proved that the two bijections are the same in this case. A reference is available on the arXiv. More generally, we have shown the Little map is the same map as the recording tableau of Edelman-Greene insertion, the RSK variant referred to above.

There is perhaps more computational evidence that the two bijections coincide (in the form of larger samples). However, to the best of my knowledge the conjecture is still open. I'd be interested to hear if you can make any progress in this direction.