Compact graph-induced semiseparable representations for two-dimensional mesh graphs

Let $M\times M$ 2D mesh graphs be graphs with the corresponding vertex sets $\mathbb{V}_\mathbb{G}$, and let a Hamiltonian path be a path $\mathbb{P}$ visiting every vertex exactly once. A GIRS-$c$ pair is a pair $(A,\mathbb{G})$ satisfying the paper's GIRS rank bound. Mesh-graph compact representation conjecture. There exists a constant $\gamma\geq 1$ such that for all $M\times M$ 2D mesh graphs $\mathbb{G}$ there exists a Hamiltonian path $\mathbb{P}$ such that, if $(A,\mathbb{G})$ is GIRS-$c$, then $(A,\mathbb{G})$ possesses a $\mathbb{G}$-SS representation with

$$r_i^g,r_i^h\leq c\gamma$$

for every $i\in\mathbb{V}_\mathbb{G}$. This is the paper's more specific conjecture for a family of graphs; the source gives no resolution.