Weak Johnson conjecture for unconditional bases

Let $X$ be a Johnson space, meaning a separable Banach space with only countably many non-isomorphic subspaces. Weak Johnson conjecture. Every Johnson space has an unconditional basis. This is one of two weaker conjectures introduced as possible approaches to Johnson's question and Ferenczi–Rosendal's ergodicity conjecture. The source does not state a resolution.