The uniqueness and ATLAS-ordering conjecture for supercharacter theories of cyclic groups of prime order

Let $p$ be a prime and let $r$ be a divisor of $p-1$. A supercharacter theory of $Z_p$ is a pair $(\mathcal{X},\mathcal{K})$ of partitions of the irreducible characters and conjugacy classes, respectively. Sort the conjugacy classes and irreducible characters of $Z_p$ according to ATLAS notations, and denote the resulting quantities by $\gamma_1(\mathcal{X})$ and $\gamma_2(\mathcal{K})$.

Uniqueness and ATLAS-ordering conjecture. For each divisor $r$ of $p-1$, there exists exactly one supercharacter theory $(\mathcal{X},\mathcal{K})$ of $Z_p$ such that every non-trivial part of each of $\mathcal{X}$ and $\mathcal{K}$ has size $r$. Moreover,

$$\gamma_1(\mathcal{X})=\gamma_2(\mathcal{K}).$$

The claim is motivated by the computed supercharacter theories of cyclic groups of prime order and by the stated count of such theories. The supplied text gives no proof or resolution.