Uniqueness of the UV-regularized information current for the BDNK bulk-viscosity model

Prove that the UV-regularized information current given in equation (37) for the BDNK-type first-order model of bulk viscosity at zero chemical potential is unique under the requirements that T∂μEμ + Tσ = 0 holds exactly along all solutions of the linearized equations (34), that Eμ is timelike future-directed, and that Tσ is nonnegative.

Background

In Section V, the authors construct a UV-regularized information current for a BDNK-type first-order model describing bulk viscosity and show that it satisfies the required thermodynamic and dynamical properties. They suggest that this construction may be the only viable one for that model.

However, they explicitly note that they currently lack a proof of uniqueness. Establishing uniqueness would clarify the degree of freedom in defining the information current for this class of models and cement the theoretical foundations of the approach.