Improving the codimension-two bound below 1/d

Determine whether, for each fixed d≥3, the codimension-two local spectral bound 1/d in the explicit weighted d-dimensional abelian Cayley complexes can be reduced while retaining polynomial Cayley degree.

Background

Theorem main-higher-dimensional provides weighted d-dimensional Cayley complexes with linear, hence optimal-order, Cayley degree at the local spectral bound 1/d. The discussion asks whether this bound can be improved without giving up polynomial degree.

The paper notes that the cube graphs used in the dimension-lifting construction have a codimension-two link with eigenvalue 1/d, and that merely increasing the dimension of the product complex before taking the skeleton cannot lower this obstruction.