Ferenczi–Rosendal conjecture on ergodicity of non-Hilbertian Banach spaces

Let $X$ be a separable Banach space that is not isomorphic to a Hilbert space. Recall that $X$ is ergodic if the equivalence relation $\mathbf{E}_0$ is Borel-reducible to the isomorphism relation on $\operatorname{Sub}(X)$, the space of subspaces of $X$. Ferenczi–Rosendal conjecture. Every separable non-Hilbertian Banach space is ergodic. This conjecture predicts that non-Hilbertian separable Banach spaces have highly complex subspace-isomorphism relations and therefore continuum many pairwise non-isomorphic subspaces. It remains open; in contrast, $\ell_2$ is non-ergodic.