Nivat's conjecture on low rectangular complexity

Let $\mathcal A$ be a finite alphabet with at least two elements, and let $\eta\in\mathcal A^{\mathbb Z^2}$ be a two-dimensional configuration. For $k,n\in\mathbb N$, let $P_\eta(k,n)$ denote the number of distinct $k\times n$ rectangular patterns occurring in $\eta$; the configuration is periodic if there is a non-zero $h\in\mathbb Z^2$ such that $\eta_{g+h}=\eta_g$ for every $g\in\mathbb Z^2$. Nivat's conjecture. If

$$P_\eta(k,n)\le kn$$

for some $k,n\in\mathbb N$, then $\eta$ is periodic. This is a two-dimensional analogue of the Morse–Hedlund theorem, which characterizes one-dimensional periodicity through low complexity. The conjecture remains open despite substantial progress.