Gowers' dichotomy theorem asserts that every Banach space either contains an HI subspace or a subspace having an unconditional basis. There are examples of HI spaces without quotients having unconditional bases (was Argyros the first who proved that?). This strange phenomenon tempts me to ask 


whether every reflexive non-HI space has a quotient with an unconditional basis? 


My apologies if this is well-known.



Quotients with unconditional bases
Every quotient of the original Gowers-Maurey space is HI; Ferenczi proved this. Argyros-Felouzis produced examples of reflexive spaces that are not HI (e.g. contain some $\ell\_p$) and have HI duals. Any of these spaces give an example of a space such that no quotient has an unconditional basis. I think it's safe to say that very little in the theory of HI spaces can be classified as 'well-known'.


