Exceptional-configuration conjecture for the discrete higher-dimensional FUP

Let $\mathcal A,\mathcal B\subset\{0,\ldots,M-1\}^2$ be digit sets, let $\mathcal C_{k,\mathcal A}$ and $\mathcal C_{k,\mathcal B}$ be the associated discrete product Cantor sets, and let $\mathcal F_{N\times N}$ denote the discrete two-dimensional Fourier transform. Exceptional-configuration conjecture. The bound

$$\|\mathbf 1_{\mathcal C_{k,\mathcal A}}\mathcal F_{N\times N}\mathbf 1_{\mathcal C_{k,\mathcal B}}\|_{\mathbb C^{N\times N}\to\mathbb C^{N\times N}}=\mathcal O(N^{-\beta})$$

holds for some $\beta>0$, unless either one digit set contains a horizontal line and the other contains a vertical line, or, for every $k$, one associated Cantor set contains a diagonal line and the other contains an antidiagonal line. The stated alternatives are the proposed obstructions to any positive FUP exponent; determining whether they are exhaustive is left open in the source.