On antichains of spreading models of Banach spaces

We show that for every separable Banach space $X$, either $\text{spw}(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\text{spw}(X)$ contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari.