On the unconditional subsequence property

We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence (fi) in ?₂(∑Lpi), where pi↓1, so that for all ε>0 and 1<q?2, every subsequence of (fi) admits a block basis (1+ε)-equivalent to the Haar basis for Lq. We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so.