The hyperfiniteness conjecture for countable amenable Borel equivalence relations

A Borel equivalence relation $E\subset X\times X$ is countable if every equivalence class is countable. It is hyperfinite if there exist finite Borel equivalence relations $E_1\subset E_2\subset\dots$ such that

$$\bigcup_{n=1}^{\infty}E_n=E.$$

It is amenable if there exist Borel functions $p_n:E\to[0,1]$ such that, for every $x\in X$ and $n\geq 1$,

$$\sum_{z\equiv_E x}p_n(x,z)=1,$$

and, for every pair $x\equiv_E y$,

$$\lim_{n\to\infty}\sum_{z\equiv_E x}|p_n(x,z)-p_n(y,z)|=0.$$

The hyperfiniteness conjecture. Every countable amenable Borel equivalence relation is hyperfinite.

Hyperfiniteness implies amenability, so the conjecture asks whether amenability is sufficient for hyperfiniteness in Borel combinatorics. The supplied text identifies this as a classical conjecture and gives no evidence of resolution.