I wonder if the following assertion in true:

Conjecture. Let $X,Y,Z$ 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 Banach space $X$, $X\oplus X$ is c.f.r. in $X$.

(B) For every 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 Fiegel and later improved by Gowers). But perhaps it is much simplier.

Thanks in advance.

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.