Prove the finite-temperature topological mutual information conjecture for string-net models

Prove the validity, for arbitrary finite temperature T, of the conjectural formula I_topo(T) = - ∑_{A ∈ Z(C)} ⟨P_A(L_c)⟩ ln[⟨P_A(L_c)⟩ 𝔇^2 / d_A^2] (Eq. (6.5)) for the topological mutual information in two-dimensional string-net models, where ⟨P_A(L_c)⟩ denotes the thermal expectation of the projector measuring flux A through a contractible loop L_c, d_A is the quantum dimension of A, and 𝔇 is the total quantum dimension of Z(C).

Background

In Section 6.1, the authors adopt a conjecture originally formulated for the Kitaev quantum double models to compute the topological mutual information at finite temperature in the string-net setting. The conjectured expression (Eq. (6.5)) relates the topological mutual information to thermal expectations of projectors onto anyon flux sectors and is motivated by general information-theoretic considerations (Kullback-Leibler divergence).

While the authors assume this conjecture holds for string-net models and show it reproduces the exact zero-temperature and infinite-temperature limits, a rigorous proof covering arbitrary finite temperatures is not provided and is explicitly flagged as outstanding. Establishing the conjecture in this broader context would underpin the finite-temperature entanglement analysis presented and clarify the universality of the scaling behavior they observe.