Near-quadratic Elekes–Rónyai expander conjecture over the reals

Let $f\in\mathbb R[x,y]$ be neither additive nor multiplicative. For a finite set $A\subset\mathbb R$, write $f(A,A)=\{f(a_1,a_2):a_1,a_2\in A\}$. Near-quadratic Elekes–Rónyai expander conjecture. For every $\epsilon>0$, there exists a constant $C_{f,\epsilon}>0$ such that

$$\lvert f(A,A)\rvert\geq C_{f,\epsilon}\lvert A\rvert^{2-\epsilon}$$

for every finite set $A\subset\mathbb R$. The conjecture proposes a near-quadratic expansion bound for nonspecial real bivariate polynomials; the source states that it is disproved by a counterexample, so the asserted universal bound is false.