TITLE: Quotients with unconditional bases
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: The space of Gowers and Maurey is reflexive and H.I. , and as proved by Ferenczi every of its quotients and its dual is also  H.I. (as  mentioned in the answers of Bill Johnson (giving the reference) and Kevin Beanland). The results of Argyros and Felouzis are answering to the question.
Moreover the remark  of Bill Johnson is to the point. S. Argyros and myself  provided a reflexive and unconditionally saturated Banach space $X$  with an H.I. dual $X^*$. Thus every quotient of   $X$ is indecomposable. Additionally we show that every quotient of $X$ has a  further quotient which is H.I.   
S. Argyros,  A. Tolias,  Indecomposability and unconditionality in duality. Geom. Funct. Anal. 14 (2004), no. 2, 247–282.