Finite-translate perfect-squares conjecture

Let $\textsc{Squares}$ denote the set of perfect squares in $\mathbb Z$. Finite-translate perfect-squares conjecture. There exists a positive integer $k$ such that there are no sets $X,Y\subset\mathbb Z$ with $|X|=|Y|=k$ satisfying

$$x+y\in\textsc{Squares}\quad\text{for all }x\in X\text{ and }y\in Y.$$

This asks whether the maximum number of translates of a fixed-size set of integers into the perfect squares is finite for some size. Such a bound would have consequences for sum-product exponents on sparse graphs; the problem is stated as open.