By the work of E. Odell and Th. Schlumprecht, we know that the separable Hilbert space $\ell_2$ is arbitrarily distortable. But I don't know if an "asymptotic" version of their result is true. To state my question, let me introduce the following terminology:
Let $(X, \| \cdot \|)$ be a separable Banach space with a normalized Schauder basis $(e_n)$ and $C \geq 1$. Let us say that $X$ is asymptotically non-distortable with constant $C$ (and with respect to the basis $(e_n)$ of $X$) if for every equivalent norm $| \cdot |$ on $X$ there exists a semi-normalized block sequence $(v_n)$ of $(e_n)$ such that for every $k$, every $k \leq n_1 < ... < n_k$ and every pair $x$ and $y$ of vectors in the span of $\{ v_{n_1}, ..., v_{n_k} \}$ with $\|x\| = \|y\| = 1$ we have $|x| / |y| \leq C$.
Question: does there exist $C\geq 1$ such that the separable Hilbert space $\ell_2$ is asymptotically non-distortable with constant $C$ and with respect to its standard unit vector basis? IF this is true, then can we take $C$ to be $1+\epsilon$ for every $\epsilon > 0$?
Of course, a similar question can be asked for a general Banach space with a Schauder basis. I think that every asymptotic $\ell_1$ space is asymptotically non-distortable for some $C\geq 1$, and for Tsirelson's space we can take $C$ to be $2+\epsilon$ for every $\epsilon > 0$. Let me recall that there exist arbitrarily distortable asymptotic $\ell_1$ spaces.
