The contractive-grid Cauchy-sequence conjecture

Let $n$ be a positive integer and let $\lambda$ satisfy $0\leq\lambda<1$. An $n$-dimensional $\lambda$-contractive grid is a pseudometric space $(\mathbb{N}_0^n,d)$ such that for every $x,y\in\mathbb{N}_0^n$, some $i\in[n]$ satisfies

$$d(x+e_i,y+e_i)\leq\lambda d(x,y).$$

The contractive-grid conjecture. Every such grid contains a 1-way Cauchy sequence. The conjecture is introduced as a formulation capturing the essence of Austin's common fixed point conjecture; the paper proves only special cases and indicates that a complete proof remains to be found.