Nivat's nonexpansive-subspace reformulation

Let $\eta\in\mathcal{A}^{\mathbb{Z}^2}$, let $X_{\eta}:=\overline{\mathcal{O}(\eta)}$ be its orbit closure under the translation action, and let $P_{\eta}(n,k)$ be the number of distinct $n$-by-$k$ rectangular patterns occurring in $\eta$. A $1$-dimensional subspace of $\mathbb{R}^2$ is nonexpansive if it is not expansive for this $\mathbb{Z}^2$-action. Nivat's reformulation. If there exist $n,k\in\mathbb{N}$ such that

$$P_{\eta}(n,k)\leq nk,$$

then there is at most one nonexpansive $1$-dimensional subspace for the translation action on $X_{\eta}$. This reformulation is motivated by the paper's theorem that, under the same complexity bound, a unique nonexpansive line forces $\eta$ to be periodic but not doubly periodic. The paper does not resolve the reformulation.