Unbounded rank super-expanders conjecture

A super-expander is a sequence of finite Cayley graphs whose expansion persists with respect to every uniformly convex Banach space, with suitable finite generating sets. Let $\mathrm{SL}(n,\mathbb{F}_p)$ denote the special linear group over the finite field with $p$ elements, and let $(\mathcal{B}_{\mathrm{uc}})$ denote the class of uniformly convex Banach spaces. Unbounded rank super-expanders conjecture. (1) For every prime $p$, the sequence $(\mathrm{SL}(n,\mathbb{F}_p))_{n\geq 4}$ can form super-expanders. (2) For every sequence of primes $(p_n)_n$, $(\mathrm{SL}(n,\mathbb{F}_{p_n}))_{n\geq 4}$ can form super-expanders. (3) The family of all simple groups of Lie type and rank at least $3$ has a mother group with property $\mathrm{(F}_{\mathcal{B}_{\mathrm{uc}}}\mathrm{)}$. The conjecture concerns the existence of super-expanders in unbounded rank and a fixed-point formulation for families of finite simple groups of Lie type; the paper reduces the special-linear-group case to relative property $(\mathrm{T})$ with respect to uniformly convex Banach spaces, but the conjectured assertions remain unresolved.