The answer to the question is "yes", allowing for a generous interpretation of "direct way".   This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group <a href="http://front.math.ucdavis.edu/1212.0791">here</a>.

See the Update to my MO question <a href="http://mathoverflow.net/questions/112734/">here</a>, 
which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative.  (This may be one of the sources you are referring to.) 

At first I had overlooked this type of answer to my own question.  Jim Carrell may want to elaborate further.  (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.)