Uncertainty conjecture for supports in the finite affine plane

Let $p$ be prime, let $f\in L(\mathbb F_p^2)$ be nonzero, and write

$$S:=\operatorname{supp} f,\qquad X:=\operatorname{supp}{\hat f}.$$

For an integer $k\in[1,p]$, let $S\subseteq\mathbb F_p^2$ and $X\subseteq\widehat{\mathbb F_p^2}$.

Finite-plane uncertainty conjecture. One has

$$\frac1k\min\{|S|,|X|\}+\frac1{p+1-k}\max\{|S|,|X|\}\ge p+1,$$

unless at least one of $S$ and $X$ is a dense subset of a union of a small number of proper cosets of the corresponding group. Perhaps it suffices to assume that neither $S$ nor $X$ can be covered by fewer than $\min\{k,p+1-k\}$ cosets.

This conjecture aims to strengthen Meshulam's uncertainty estimate for the elementary abelian group of rank $2$, distinguishing the exceptional configurations associated with unions of proper cosets. The source presents it as a conjecture and gives no resolution.