Uniqueness of unconditional bases in nonlocally convex ℓ1-products

We show that the ℓ1-product (X⊕X⊕⋯⊕X⊕⋯)1 has a unique unconditional basis up to permutation for a wide class of nonlocally convex quasi-Banach spaces X, even without knowing whether X has a unique unconditional basis or not.