While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform structures [Johson&Lindenstrauss&Schechtman]. [Ribe] pointed out that two non-isomorphic spaces may have homeomorphic uniform structures (See [Benyamini&Lindenstrauss, Chap 10, Sec 10.4]).


The classical nonunique example in [Benyamini&Lindenstrauss, Chap 10, Prop 10.32-10.34] shown that p-convexified Tsirelson spaces $\mathcal{T}^{(p)}$ for $1<p<\infty$ admit *exactly two* different uniform structures. 


**Is there an example that a Banach space $X$ admits more than two uniform structures?**


[Johson&Lindenstrauss&Schechtman]Johson, W. B., Joram Lindenstrauss, and Gideon Schechtman. "Banach spaces determined by their uniform structures." Geometric & Functional Analysis GAFA 6.3 (1996): 430-470, doi: [10.1007/BF02249259](http://dx.doi.org/10.1007/BF02249259), [arXiv:math/9701203](https://arxiv.org/abs/math/9701203).


[Benyamini&Lindenstrauss]Benyamini, Yoav, and Joram Lindenstrauss. Geometric nonlinear functional analysis. Vol. 48. American Mathematical Soc., 1998.


[Ribe]Ribe, Martin. "On uniformly homeomorphic normed spaces." Arkiv för matematik 14.1-2 (1976): 237-244, doi: [10.1007/BF02385837](http://dx.doi.org/10.1007/BF02385837).



Example of Banach spaces with non-unique uniform structures
In Theorem 5.8 of my paper with Lindenstrauss and Schechtman that you reference we show that for every $n=0,1,2,\dots$ there exists a Banach space that is uniformly homeomorphic to exactly $2^n$ mutually non isomorphic spaces. AFAIK, it is still open whether $2^n$ can be replace by any number that is not a power of $2$. See (5) at the end of the paper.


