Jena–Genin–Mosca conjecture on Clifford Pauli partitioning

Let $q$ be prime, let $\mathcal{S}\subseteq \pauli{q}[*]$ be a set of generalized Pauli operators, and let $\mathcal{C}$ denote the generalized Clifford group. A partition is a collection of parts whose operators are simultaneously diagonalizable by elements of the specified gate set. Jena–Genin–Mosca conjecture. For almost all sets $\mathcal{S}$, the number of parts in a minimal partition of $\mathcal{S}$ with respect to $\mathcal{C}$ is bounded above by

$$\left( \frac{1}{2} + o(1) \right) \left( \log_2(q) + o(1) \right) \frac{|\mathcal{S}|}{\log_2(|\mathcal{S}|)}.$$

This conjecture extends the preceding estimate from linearly independent sets to almost all sets, with linear dependence expected to produce larger co-cliques and hence require no more colors. The source gives no resolution, so the conjecture remains open.