2 problems

Power-weight duality conjecture. The power weights $w^{(\ell)}$, $\ell \geq 2$, do not respect duality over any $\mathbb{Z}/m\mathbb{Z}$ with $m \geq 4$.

Dual-partition refinement conjecture. If $|H_i|\geqslant2$ for all $i\in\Omega$, then the dual partition of $\mathcal{Q}(\boldsymbol{H},\boldsymbol{P})$ is finer than…