Erdős–Turán conjecture for irreducible-in-variables two-variable polynomials

Let $f\in\mathbb Z[x,y]$ be a two-variable polynomial that is not decomposable as

$$f(x,y)=g(x)h(y),$$

where $g,h\in\mathbb Z[x]$. Let $\mathcal A,\mathcal B\subseteq\mathbb Z^+$ be finite sets with $|\mathcal A|\geq|\mathcal B|\geq2$, and let $\omega_{\mathbb N}(n)$ denote the number of distinct prime divisors of the positive integer $n$.

Erdős–Turán conjecture for two-variable polynomials. There exists a constant $c$ depending only on $f$ such that

$$\omega_{\mathbb N}\left(\prod_{a\in\mathcal A,\ b\in\mathcal B}f(a,b)\right)\geq c\log|\mathcal A|.$$

This conjecture seeks to generalize the Erdős–Turán theorem from products of sums to products of values of arbitrary two-variable polynomials that do not separate as a product of a polynomial in $x$ and a polynomial in $y$. Its status is not resolved in the supplied source context.