Inverse inequality conjecture for quantum uniformity and hierarchy-overlap measures

Let $U$ be an $n$-qudit unitary, and let $\left|\left|U\right|\right|_{Q^k}$ and $\left|\left|U\right|\right|_{q^k}$ denote its quantum uniformity norm and hierarchy-overlap measure, respectively. Inverse inequality conjecture. If

$$0<c\leqslant\left|\left|U\right|\right|_{Q^{k}},$$

then there exists a constant $d$, independent of $n$, such that

$$0<d\leqslant\left|\left|U\right|\right|_{q^{k}}.$$

The preceding direct inequality shows that the hierarchy-overlap measure is bounded above by the quantum uniformity norm; this conjecture asks for a dimension-independent converse lower bound. Its resolution would establish quantitative equivalence between the two measures beyond their shared characterization of the Clifford hierarchy.