Borel almost finiteness conjecture for uniformly Borel amenable Følner graphs

Let $G$ be a bounded-degree Borel graph. Call $G$ a uniformly Borel amenable Følner graph when it has the uniform Borel amenability and Følner properties defined in the source. Call $G$ Borel almost finite if, for every $\varepsilon>0$, there is $k>0$ such that $G$ admits an $(\varepsilon,k)$-Følner tiling.

Borel Almost Finiteness Conjecture. Every uniformly Borel amenable Følner graph is Borel almost finite.

This conjecture is a guiding principle for uniformly Borel amenable Følner graphs and was originally proposed by Marks for free amenable actions. Its converse is false: there are Borel almost finite graphs that are not Borel amenable.