Modified Nivat conjecture in terms of nonexpansive subspaces

Let $\eta\in\mathcal{A}^{\mathbb{Z}^2}$, let $X_{\eta}:=\overline{\mathcal{O}(\eta)}$ be its translation orbit closure, and for a finite set $\mathcal{S}\subseteq\mathbb{Z}^2$ define the discrepancy by

$$D_{\eta}(\mathcal{S}):=P_{\eta}(\mathcal{S})-|\mathcal{S}|,$$

where $P_{\eta}(\mathcal{S})$ is the number of distinct $\eta$-colorings of $\mathcal{S}$. A $1$-dimensional subspace is nonexpansive if it is not expansive for the translation action. Modified Nivat conjecture. If there exists an $n$ by $k$ rectangular subset $R\subseteq\mathbb{Z}^2$ satisfying

$$D_{\eta}(R)\leq 0,$$

then there is at most one nonexpansive $1$-dimensional subspace for the translation action on $X_{\eta}$. This is a reformulation using discrepancy rather than rectangular complexity; the paper leaves it open.