Weakly null sequences in $L_1$

We construct a weakly null normalized sequence $\{f_i\}_{i=1}^{\infty}$ in $L_1$ so that for each $\varepsilon>0$, the Haar basis is $(1+\varepsilon)$-equivalent to a block basis of every subsequence of $\{f_i\}_{i=1}^{\infty}$. In particular, the sequence $\{f_i\}_{i=1}^{\infty}$ has no unconditionally basic subsequence. This answers a question raised by Bernard Maurey and H. P. Rosenthal in 1977. A similar example is given in an appropriate class of rearrangement invariant function spaces.