Naor's finite analogue of the Cornulier–Tessera–Valette conjecture

Let $\{G_n\}_n$ be a sequence of finite groups with $\sup_n \operatorname{rank}(G_n)<\infty$. For a generating set $S$ of size $r$, let $d_S$ be the associated word metric, and let $c_2(G_n,d_S)$ denote its infimal bi-Lipschitz distortion into Hilbert space. For $S\in {G_n\choose r}$, write $\langle S\rangle=G_n$. Naor's finite conjecture. The following five conditions are equivalent:

1. For all $r>\sup_n \operatorname{rank}(G_n)$,

$$\sup_n\sup_{S\in {G_n\choose r}:\langle S\rangle=G_n}c_2(G_n,d_S)<\infty.$$

2. There exists $r\in\mathbb{N}$ such that

$$\sup_n\sup_{S\in {G_n\choose r}:\langle S\rangle=G_n}c_2(G_n,d_S)<\infty.$$

3. For all $r>\sup_n \operatorname{rank}(G_n)$, there are generating sets $S_n$ of size $r$ such that

$$\sup_n\inf_{S\in {G_n\choose r}:\langle S\rangle=G_n}c_2(G_n,d_S)<\infty.$$

4. There exists $r\in\mathbb{N}$ and generating sets $S_n$ of size $r$ such that

$$\sup_n\inf_{S\in {G_n\choose r}:\langle S\rangle=G_n}c_2(G_n,d_S)<\infty.$$

5. There exists $r\in\mathbb{N}$ and, for each $G_n$, an abelian subgroup $H_n\leq G_n$ of rank at most $r$ such that

$$\sup_n[G_n:H_n]<\infty.$$

The conjecture is a finite-group analogue of the assertion that Hilbert-space embeddability forces virtual abelianness. Its status is not resolved in the supplied source context.