TITLE: A minimality problem for a class of Banach spaces
QUESTION [6 upvotes]: The following question is related to the previous question Minimality properties of James' space; I post it as a new question since the system does not allow me to add a comment.
Question  Consider the following class of non-Hilbertian spaces:
    $X_{p,2}=(\sum_{n=1}^\infty \oplus\ell^p_n)_2$, $1\le p\le \infty$, $p\neq 2$.
 Is it true that the only infinite dimensional Banach space that is isomorphically embedded into anyone of them is the Hilbert space?
Notice that all these spaces are subspaces of the space $\mathcal{J}$.

REPLY [2 votes]: I think such a space has type 2 and cotype 2 so by Kwapien's theorem it is isomorphic to a Hilbert space.