Finite-commutative-ring product-of-difference-sets conjecture

Let $\alpha>0$. A finite commutative ring is a ring $R$ with finitely many elements, and for a set $E\subset R$ write

$$(E-E)\cdot(E-E)=\{(e_1-e_2)(e_3-e_4)\mid e_1,e_2,e_3,e_4\in E\}.$$

Finite-ring product-of-difference-sets conjecture. There exist $N$ and $k$, depending only on $\alpha$, such that for every finite commutative ring $R$ with $|R|\ge N$ and every set $E\subset R$ satisfying $|E|\ge\alpha|R|$, the set $(E-E)\cdot(E-E)$ contains a subring $R_0$ such that

$$|R|/|R_0|\le k.$$

This is proposed as a quantitative analogue of the paper's results for integer and finite cyclic settings. The source presents it as a belief for finite commutative rings and gives no resolution.