The conjecture which you stated is false. A counterexample is contained in the proof of Figiel[ \[*Studia Math.* **42** (1972), 295–306\]][1]. 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. 


  [1]: http://matwbn.icm.edu.pl/ksiazki/sm/sm42/sm42125.pdf