The following question is related to the previous 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 $J$.

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 $\mathcal{J}$.