The block-sequence permutative-equivalence conjecture

Let $X$ be a Banach space with a basis, respectively an unconditional basis, which is not equivalent to the canonical basis of $c_0$ or $\ell_p$ for $1\leq p<+\infty$. A normalized block-sequence is a block-sequence whose vectors have norm one, and in the unconditional case a sequence of disjointly supported blocks consists of finitely supported vectors with pairwise disjoint supports. Let $E_0$ denote eventual equality on $2^{\mathbb N}$. The block-sequence permutative-equivalence conjecture. Then $E_0$ is Borel reducible to permutative equivalence between normalized block-sequences, respectively sequences of disjointly supported blocks, of $X$.

The claim concerns the complexity of permutative equivalence, which is already analytic non-Borel in the relevant setting. It predicts that every basis outside the classical $c_0$ and $\ell_p$ cases contains sufficiently many pairwise complicated block sequences.