I am interested in the following question about James' quasi-reflexive Banach space $\mathcal{J}$:

Does there exists a non-Hilbertian subspace $X$ of $\mathcal{J}$ such that $X$ isomorphically embeds into every non-Hilbertian subspace of itself?

Here, by "subspace" I mean "closed, infinite-dimensional vector subspace", and by "Hilbertian" I mean "isomorphic to $\ell_2$".

I vaguely recall having found, one year ago, a paper proving that the answer to this question was no, or at least giving a similar/partial result suggesting that the answer should be no. Problem is, I don't manage to find this paper again, I don't even remember who were the authors and what was the exact result they proved. Do some of you recall having seen something like that?