The higher-dimensional spacetime density conjecture

For an integer $m\geq 2$ and $I\subseteq\mathbb{Z}^m$, let its upper density be

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

Higher-dimensional spacetime density conjecture. There exists an integer $m\geq 2$ such that, for every $\epsilon>0$, there exists a positive integer $d$ with the property that whenever $I\subseteq\mathbb{Z}^m$ has upper density greater than $\epsilon$, it contains points $(x_1,\ldots,x_m)$ and $(x_1',\ldots,x_m')$ satisfying

$$(x_1-x_1')^2-\sum_{j=2}^m(x_j-x_j')^2=d^2.$$

This is explicitly presented as a weakening of the two-dimensional density conjecture when more temporal dimensions are allowed, and it remains open.