TITLE: Is X+X finitely representable in X?
QUESTION [5 upvotes]: I wonder if the following assertion in true:
Conjecture. Let $X,Y,Z$ be infinite-dimensional Banach spaces such that both $Y$ and $Z$ are crudely finitely representable (c.f.r. for short) in $X$. Then $Y\oplus Z$ is c.f.r. in $X$.
Remark some equivalent formulations of the above conjecture.
(A) For every infinite-dimensional Banach space $X$, $X\oplus X$ is c.f.r. in $X$.
(B) For every infinite-dimensional Banach space $X$, $X\oplus X$ is isomorphic to a subspace of an ultrapower of $X$.
I guess  this question could be  connected with  those of whether a Banach space isomorphic to its square (solved in the negative by Figiel and later improved by Gowers).  But perhaps it is much simpler.
Thanks in advance.

REPLY [7 votes]: The conjecture which you stated is false. A counterexample is contained in the proof of Figiel [Studia Math. 42 (1972), 295–306]. He actually proves that squares of finite-dimensional subspaces of the space he constructs are not uniformly embeddable into the space itself.
I am unaware of a simpler counterexample for infinite-dimensional spaces.