Is there an injective cubic polynomial $\mathbb Z^2\rightarrow\mathbb Z$?
This question asks exactly what its title suggests: is there a polynomial of degree $3$ in $\mathbb Z[x,y]$ whose induced map $\mathbb Z^2\rightarrow\mathbb Z$ is injective? The question is trivial for polynomials of degree $4$ (where the answer is yes) or polynomials of degree $2$ or less (where the answer is no).
I recognize that this may not be wholly in the spirit of the thread, since I have suspicions that the question is open given the existence of two unanswered questions on MO asking about polynomial bijections (one on $\mathbb Q^2$ and one on $\mathbb Z^2$). However, I suspect that my question is considerably easier, as it asks only about injectivity and only about cubics (where, from my amateur knowledge, it seems like more tools are available than for general polynomials). In any case, I note that there has been some amount of research regarding both the linked questions, and think that it's not unlikely that an expert in the field could have some references or insights to contribute to the question.