Classification conjecture for time-frequency representations on finite Abelian groups

Let $G$ be a finite Abelian group, let $V_g f$ denote the time-frequency representation appearing in the paper, let $\widehat f$ denote the Fourier transform of $f$, and write $\\|f\\|_0$ for the cardinality of the support of $f$. Then, for almost every $g\in\mathbb{C}^G$, the classification conjecture asserts

$$\big\{(\\|f\\|_0, \\|V_g f\\|_0),\ f\in\mathbb{C}^G\setminus\{0\}\big\}=\big\{(\\|f\\|_0, \\|\widehat f\\|_0+|G|^2-|G|),\ f\in\mathbb{C}^G\setminus\{0\}\big\}.$$

This would extend the prime-order classification of the pairs $(\\|f\\|_0,\\|V_g f\\|_0)$ to arbitrary finite Abelian groups and describe the generic support-size behavior of the time-frequency representation.