On uniform and coarse rigidity of $L^p([0,1])$

If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1\leqslant p<\infty$) and $X$ admits a uniformly continuous map $X\oversetϕ\longrightarrow E$ into a Banach space $E$ satisfying $\inf_{\|x-y\|=r} \| ϕ(x)-ϕ(y)\|>0$ for some $r>0$, then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.

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