Removing hyper-regularity from HDX degree lower bounds

Determine whether the hyper-regularity assumption can be removed from the super-exponential degree lower bound proved for skeletons of Trickling-Down high dimensional expanders; specifically, prove analogous degree lower bounds for non-hyper-regular complexes (e.g., Ramanujan complexes) or develop a finer concentration framework that implies such bounds.

Background

The super-exponential degree lower bounds established in the paper currently require a hyper-regularity condition on the complex. Many prominent HDX constructions, such as Ramanujan complexes, are not hyper-regular and have highly unbalanced links, preventing direct application of the result.

The authors conjecture that refined concentration tools or their applications could eliminate this assumption and extend the lower bounds to broader HDX families.