Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$

Question

Let $X$ be a Banach space with basis $(e_n)_{n=1}^\infty$, and suppose that $(x_i)_{i=1}^\infty$ is a normalized block basic sequence of $(e_n)_{n=1}^\infty$. In addition assume that $(x_i)_{i=1}^\infty$ is subsymmetric, weakly null, and such that $\sup_i \left\vert \text{supp}(x_i) \right\vert = \infty$.

My question: is it possible to extract a subsequence $(x_{n_j})_{j=1}^\infty$ for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$? Perhaps find an equivalent normalized block basic sequence $(y_j)_{j=1}^\infty$ of $(e_n)_{n=1}^\infty$ for which $\sup_j \left\vert \text{supp}(y_j) \right\vert < \infty$?

Answer 1

No, take, for instance, a reflexive Orlicz sequence space $\ell_M$ not isomorphic to $\ell_p$ and a block sequence $(x_i)$ equivalent to $\ell_p$ basis that space contains. Every block sequence in the space with uniform bound on the support will be equivalent to the unit vector basis of $\ell_M$.