The Bourgain ℓ1-index of mixed Tsirelson space

Suppose that (Fn)n=0∞ is a sequence of regular families of finite subsets of N such that F0 contains all singletons, and (θn)n=1∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T(F0,(θn,Fn)n=1∞) is the completion of c00 with respect to the implicitly defined norm||x||=max||x||F0,supn∈N sup θn∑i=1k||Eix||,where ||x||F0=supF∈F0||Fx||ℓ1 and the last supremum is taken over all sequences (Ei)i=1k in [N]<∞ such that max Ei