Let $f(x, y), g(x, y) \in \mathbb{C}[x, y]$. Consider the image

$$S = (f(\mathbb{C}^2), g(\mathbb{C}^2)).$$

If for an integer $d \in \mathbb{N}$ the set $S$ contains a set $A_1 \times A_2, A_1, A_2 \subset \mathbb{C}$ being such that the minimum of the cardinalities of $|A_1|, |A_2|$ is $d$, one can say that "$S$ contains a product set of width $d$".

If $S$ contains product sets of arbitrarily large width, then one can observe that it follows from the Combinatorial Nullstellensatz that the polynomials $f(x, y)$ and $g(x, y)$ are algebraically independent over $\mathbb{C}$.

I wonder whether it is known if the converse is true (that is, if $f(x, y)$ and $g(x, y)$ are algebraically independent over $\mathbb{C}$, then $S$ contains product sets of arbitrarily large width)?