Cantorval spectrum conjecture for a two-dimensional quasicrystal Schrödinger operator

Let $\gamma=\frac{\sqrt{5}-1}{2}$ and, for $j\in\{1,2\}$ and $k\in\mathbb{Z}$, define

$$s_j^{\lambda_j}(k)=\lambda_j\chi_{[1-\gamma,1)}(\{k\gamma\}).$$

Set $V(m,n)=s_1^{\lambda_1}(m)+s_2^{\lambda_2}(n)$ and let $H$ on $l^2(\mathbb{Z}^2)$ be given by

$$[H\psi](m,n)=\psi(m+1,n)+\psi(m-1,n)+\psi(m,n+1)+\psi(m,n-1)+V(m,n)\psi(m,n).$$

Cantorval spectrum conjecture. There exist values of $\lambda_1$ and $\lambda_2$ such that the spectrum $\sigma(H)$ is a Cantorval; that is, the spectrum is the closure of its interior and no connected component is isolated.

The conjecture concerns the expected structure of the spectrum for a separable two-dimensional quasicrystal potential. The source gives no resolution status.