The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into subspaces of dimension $2$, and also admit a Schauder basis which is the union of some natural bases of the $2$-dimensional subspaces.

**QUESTION:** Suppose that, for some $k\in\mathbb N$, the Banach space $X$ admits a symmetric FDD into subspaces of dimension $k$.

Can we assure that $X$ admits a Schauder basis?