Verstraëte's dichotomy conjecture for polynomial value sets

Let $f\in\mathbb{Z}[x]$ and let $k$ be a positive integer. For a set $A\subseteq[n]$, require that no product of $k$ distinct elements of $A$ belongs to the value set of $f$.

Verstraëte's dichotomy conjecture. For some constant $\rho=\rho(k,f)$ depending only on $k$ and $f$, the maximal size of such a set $A$ is either

$$(\rho+o(1))n\quad\text{or}\quad(\rho+o(1))\pi(n)\qquad\text{as }n\to\infty.$$

Verstraëte established the two possible orders of magnitude for certain classes of polynomials and conjectured that these are the only possibilities. The dichotomy remains open in general.