The two-dimensional threshold-generation conjecture for the Molchanov–Vainberg operator

Let $D$ be the discrete Molchanov–Vainberg Laplacian, let $\boldsymbol{\Theta}_{m,\kappa}(D)$ denote the threshold sets defined in the paper, and let $\mathbb{N}_e$ denote the even positive integers. Threshold-generation conjecture. For $d=2$ and $\kappa\geq4$ with $\kappa\in\mathbb{N}_e$,

$$\bigcup_{m\geq0}\boldsymbol{\Theta}_{m,\kappa}(D)=\boldsymbol{\Theta}_{\kappa}(D).$$

The claim is motivated only by the absence of counterexamples in the paper, and no proof or disproof is supplied.