The countability conjecture for threshold sets

Let $D$ be the discrete Molchanov–Vainberg Laplacian, let $\boldsymbol{\Theta}_{m,\kappa}(D)$ and $\boldsymbol{\Theta}_{\kappa}(D)$ be the threshold sets defined in the paper, and let $\mathbb{N}_e$ denote the even positive integers. Threshold-countability conjecture. For every even $\kappa\geq2$,

$$\bigcup_{m\geq0}\boldsymbol{\Theta}_{m,\kappa}(D)\quad\text{and}\quad\boldsymbol{\Theta}_{\kappa}(D)$$

are countable sets. The paper presents this as an open conjecture while listing related questions about accumulation points and density.