Let $X$ be a Banach space and $T \colon \ell^2\rightarrow \ell^2$ be a bounded linear map. Suppose that the linear map $T\otimes Id_ {X}:\ell^2\otimes X\rightarrow \ell^2\otimes X$ which maps $e_i \otimes x$ to $e_i\otimes T(x) $ is not bounded when we use the norm on $\ell^2\otimes X$ induced by the Banach space $\ell^2(X)$.
Does it exist a sequence $(X_n)$ of finite dimensional subspaces of $X$ such that $ ||T \otimes Id_{X_n}||_{ \ell^2(X_n)\to \ell^2(X_n) } \xrightarrow[n \to +\infty]{}+\infty\ ? $