Quasistability in the unresolved higher-dimensional regime

Establish whether quasistability holds for $(\gamma,a)$-minimal matchings in higher dimensions when $0<\gamma<1$ and $\log(\overline{a}/\underline{a})>1-\gamma$, where $\underline{a}$ and $\overline{a}$ are respectively the infimum and supremum of the directional coefficient on the unit sphere.

Background

The paper proves quasistability for unbalanced minimal matchings in one dimension and, in higher dimensions, for all exponents outside the interval $(0,1)$. For $0<\gamma<1$, the higher-dimensional argument yields quasistability only under a quantitative restriction on the variation of the directional coefficient $a(\theta)$.

The remark isolates the complementary parameter regime in which the available estimate does not establish quasistability. Determining whether quasistability actually fails or can nevertheless be proved there remains unresolved and is relevant to extending the comparison theorem between balanced and unbalanced matchings to higher dimensions.