A Banach space $X$ is said to have Fourier type $p\in[1,2]$ if the Fourier transform $\hat{f}(s):=\int_{\mathbb{R}}e^{-ist}f(t)dt$ defines a bounded linear operator from $L_{p}(\mathbb{R},X)$ to $L_{p'}(\mathbb{R},X)$ where $p'$ is the Holder conjugate of $p$.
It is well-known that $X$ is a Hilbert space iff it has Fourier type $p=2$, whereas every Banach space has Fourier type $p=1$. See this paper by Girardi and Weis, Definition 2.2.
It is natural to ask whether or not various Hilbert-like spaces have Fourier type $p$ for some $p\in(1,2)$. In particular, my question is: must $X$ have Fourier type $p$ for some $p\in(1,2)$ if $X$ is asymptotic-$\ell_{2}$ w.r.t. a normalized (conditional or unconditional) basis?
Note: $X$ is asymptotic-$\ell_{2}$ w.r.t. to a normalized basis $(b_{k})_{k}$ if there exist $C_{1},C_{2}>0$ such that for all $n\in\mathbb{N}$, \begin{equation*} C_{1}\left(\sum_{j=1}^{n}\|x_{j}\|^{2}\right)^{\frac{1}{2}}\leq\left\Vert\sum_{j=1}^{n}x_{j}\right\Vert\leq C_{2}\left(\sum_{j=1}^{n}\|x_{j}\|^{2}\right)^{\frac{1}{2}}\end{equation*} for all block sequences $(x_{j})_{j=1}^{n}$ of $(b_{k})_{k}$ with $\min\text{supp}(x_{1})\geq n$.
I suspect that the answer to my question is no based on some quick scratch work and the fact that an asymptotic-$\ell_{2}$ space need not be reflexive (and therefore need not be uniformly convex), but I would just like to verify this intuition if possible.
