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.