The following question is inspired by Existence of function $g$ such that $f(x,y)\le g(x)+g(y)$ , which has been closed for unknown reason and which may have a wellknown answer. Is the following true?

Let $X$ be an uncountable set. Then there is a function $f \colon X \times X \to \mathbb{N}$ such that for any function $g \colon X \to \mathbb{N}$ there is $(x,y) \in X^2$ with $f(x,y) > g(x) + g(y)$.

It is easy to show that this is false if $X$ is countable.