The block-subspace dichotomy and classification conjecture

Let $X$ be a Banach space with an unconditional basis. A block-subspace is a closed subspace generated by a block-sequence of that basis, and $E_0$ is the equivalence relation of eventual equality on $2^{\mathbb N}$. The block-subspace dichotomy and classification conjecture. Is it true that either $X$ is isomorphic to each of its block-subspaces or $E_0$ is Borel reducible to isomorphism between the block-subspaces of $X$? Is it true that if $X$ is isomorphic to its block-subspaces then $X$ is isomorphic to $c_0$ or $\ell_p$? Are these assertions true when block-subspaces are replaced by subspaces supported by disjointly supported blocks?

The question is motivated by the homogeneity of $c_0$ and $\ell_p$ and by the fact that many results about ergodicity can be formulated using block-subspaces. It asks both for a dichotomy in descriptive complexity and for a classification of spaces homogeneous under block-subspace formation.