I have two quick questions:

It can be shown without too much trouble (using methods from Altshuler/Casazza/Lin, 1973) that any Lorentz sequence space admits a unique (up to equivalence) subsymmetric basis (which is also symmetric).

Question 1. Are there any other known examples of a Banach space admitting a unique subsymmetric basis?

In 2004, Sari showed that the Tirilman spaces admit a subsymmetric basis but not a symmetric one.

Question 2. Are there any other known examples of a Banach space admitting a subsymmetric basis but not a symmetric one?

I am especially interested in Banach spaces satisfying both properties simultaneously---that is, a Banach space admitting a unique subsymmetric basis which is not symmetric.

Thanks!