Prove stability for all wavenumbers of the modified hyperbolization (modify-A)

Prove stability for all wavenumbers k of the hyperbolization system given by equation (modify-A) for the general linear scalar evolution equation u_t + Σ_{j=0}^{m-1} α_j ∂_x^j u + σ_0 ∂_x^m u = 0, where the signed permutation matrix P is chosen as in equation (stableP-plus).

Background

After establishing necessary and sufficient conditions for stability in special cases and proving stability for the unique hyperbolization in the constant-coefficient high-order linear model, the authors extend to the general linear scalar case and propose the system (modify-A). They show stability at k=0 and as |k|→∞ but do not provide a full stability proof for all wavenumbers.

They explicitly state that they have not found a way to generalize their main stability theorem to (modify-A) and prove stability across all wavenumbers, marking this as an unresolved point despite suggestive evidence and computational experiments.