Weak ergodicity conjecture for non-ergodic Banach spaces

Let $X$ be a separable Banach space. A space is non-ergodic when it is not ergodic, and a Banach space is Hilbertian when it is isomorphic to a Hilbert space. Weak ergodicity conjecture. Every non-ergodic non-Hilbertian separable Banach space contains a non-Hilbertian subspace having an unconditional basis. This conjecture weakens the assertion that every separable non-Hilbertian Banach space is ergodic by seeking an unconditional-basis subspace inside a potential counterexample. The source does not state a resolution.