Optimal polynomials for the Hoffman ratio-type bound

Determine the optimal degree-k polynomial for the Hoffman ratio-type eigenvalue bound on the quantum k-distance chromatic number for every k other than k = 2 and k = 3.

Background

The Hoffman ratio-type lower bound depends on selecting a polynomial that maximizes the resulting spectral estimate. Closed-form optimal polynomials are available for k = 2 and k = 3, while the paper states that optimal polynomials remain unknown for all other values of k. A linear program can optimize the bound computationally, but this does not provide the requested closed-form characterization.