$\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator{\E}{E}$

An $(r,s)$ Pythagorean biclique is two sets of natural numbers $A=\{a_1,\ldots,a_r\}$ and $B=\{b_1,\ldots,b_s\}$ such that the distance from $a_i$ to $b_j$ is an integer, for every $i\in\{1,\ldots,r\}$ and every $j\in\{1,\ldots,s\}$. Currently, the largest Pythagorean biclique is of size $(7,9)$.

Is it true that for every $k$, there exists a $(k,k)$ Pythagorean biclique?

This question was posed by David Eppstein on his blog.