Lower bounds on degree for locally unbalanced HDX

Investigate whether every infinite family of bounded-degree high dimensional expanders that are either λ-Trickling-Down or η-spectrally independent must have super-exponential degree as a function of the dimension; namely, ascertain whether deg(X_n) exp(ω(d)) for sufficiently large members X_n of the family.

Background

The paper proves super-exponential degree lower bounds for certain hyper-regular HDX and highlights a threshold phenomenon at the TD barrier. Extending these lower bounds to complexes with highly unbalanced links (e.g., Ramanujan-type constructions) would settle whether degree blow-up is inevitable.

This question captures a central limitation in using HDX as gadgets in complexity-theoretic applications, where degree controls the overhead.