In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates and identify the cells with the elements of $\mathbb{Z}^2$.
Given a positive integer $N$, we denote by $Q_N$ the square on $\mathbb{Z}^2$ with center at the origin and of size $(2 N+1) \times(2 N+1)$ : $$ Q_N:=\{(n, m):|n|,|m| \leq N\} . $$
The translation of this square by a vector $x$ with integer coordinates is denoted by $Q_N(x)$. For a set $S \subset \mathbb{Z}^2$, we denote by $|S|$ the number of elements in $S$.
Let $\varepsilon \in(0,1)$ be a positive number. We say that $|u|$ is bounded by 1 on (1- $)$ portion of the lattice $\mathbb{Z}^2$ if for some $N_0>0$ $$ \left|Q_N \cap\{|u| \leq 1\}\right| \geq(1-\varepsilon)\left|Q_N\right| $$ for all $N \geq N_0$ There exists $\varepsilon>0$ such that if $u$ is a harmonic function on $\mathbb{Z}^2$ and $|u|$ is bounded by 1 on $(1-\varepsilon)$ portion of $\mathbb{Z}^2$, then $u$ is constant.
My problem is whether the result is still true if we change the function $u$ to be a function that is bounded above(rather than bounded) on a large portion of $\mathbb{Z}^2$.