Density conjecture for integral points at spacetime distance

For a set $I\subseteq\mathbb{Z}^2$, define its upper density by

$$\delta(I)=\limsup_{R\to\infty}\frac{|I\cap[-R,R]^2|}{(2R+1)^2}.$$

Density conjecture. For every $\epsilon>0$, there exists some positive integer $d$ such that if $I\subseteq\mathbb{Z}^2$ has upper density greater than $\epsilon$, then there is a pair of points $(x,y),(x',y')\in I$ with

$$(x-x')^2-(y-y')^2=d^2.$$

This conjecture is a density strengthening of the corresponding finite-colouring problem for the spacetime quadratic form. The paper proves an analogous density result in three dimensions, while this two-dimensional statement remains open.