Hyperfiniteness of increasing unions of Borel equivalence relations

Definitions and mathematical statement

A standard Borel space is a set equipped with the Borel structure of some Polish topology. On such a space $X$, a Borel equivalence relation is a Borel subset of $X\times X$ that is an equivalence relation. It is finite when all its classes are finite, with no common bound on their sizes required. It is hyperfinite when it has an exhaustion \[ R=\bigcup_{k\in\mathbb N}R_k,\qquad R_k\subseteq R_{k+1}, \] by finite Borel equivalence relations on $X$.

Suppose $F_1\subseteq F_2\subseteq\cdots$ are hyperfinite Borel equivalence relations on the same standard Borel space $X$. Must \[ E=\bigcup_{n\in\mathbb N}F_n \] be hyperfinite? The conclusion asks for one increasing exhaustion of $E$ by finite Borel equivalence relations. All assertions concern the whole space, without discarding a null set or a meagre set.

Short English statement

Does taking an increasing countable union preserve hyperfiniteness for Borel equivalence relations?

Sources

S. Gao and M. Xiao, An order analysis of hyperfinite Borel equivalence relations, arXiv:2404.17516v2, Problem 2.9 and Section 2, https://arxiv.org/html/2404.17516v2. Adapted under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The general question is also stated as open in the introduction to Hyperfiniteness of bounded-to-one actions of commutative monoids, arXiv:2608.18439v2, https://arxiv.org/html/2608.18439v2.