Jena–Genin–Mosca conjecture on single-qudit Clifford partitioning

Let $q$ be prime, let $\mathcal{S}\subseteq \pauli{q}[*]$ be a set of generalized Pauli operators, let $sq\mathcal{C}$ denote the gate set of single-qudit Clifford operators, and let $m$ be the length of the largest Pauli operator in $\mathcal{S}$. A partition is a collection of parts whose operators are simultaneously diagonalizable by elements of the specified gate set. Jena–Genin–Mosca conjecture. Given $\mathcal{S}\subseteq \pauli{q}[*]$, the number of parts in a minimal partition of $\mathcal{S}$ with respect to $sq\mathcal{C}$ is expected to be bounded below by

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

The bound expresses the expected cost of restricting measurements to single-qudit Clifford operations, compared with arbitrary Clifford operations. The source gives no resolution, so the conjecture remains open.