fa.functional analysis - Asymptotic non-distortion of the separable Hilbert space

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 $Cgeq 1$ 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 $Cgeq 1$, 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.