Balog–Wooley decomposition conjecture

Let $A$ be a finite subset of $\mathbb{R}$. For a finite set $X$, write

$$E^+(X)=E_2^+(X),\qquad E^\times(X)=E_2^\times(X),$$

where $E_k^+(X)=\sum_x r_{X-X}^k(x)$ and $E_k^\times(X)=\sum_x r_{X/X}^k(x)$. Let $\eta\leq 2/3$ be given. Balog–Wooley conjecture. For every finite set $A\subseteq\mathbb{R}$, there are sets $B,C$ such that $A=B\sqcup C$ and

$$\max\\{E^+(B),E^\times(C)\\}\lesssim |A|^{3-\eta}.$$

This decomposition seeks to partition a set into an additively unstructured part and a multiplicatively unstructured part. The source gives no resolution status.