Necessary and sufficient conditions for the Markovian Mpemba effect

Continuous-time Markov jump process on n states with energies $E_i$ and barriers $B_{ij}$, Arrhenius rates $W_{ij} = \exp(-(B_{ij} - E_j)/T_b)$ obeying detailed balance at bath temperature $T_b$, initial condition the Boltzmann distribution $\pi(T_0)$. Expand $p(t) - \pi(T_b) = \sum_{k \ge 2} a_k(T_0) v_k \exp(-\lambda_k t)$ with $0 < \lambda_2 < \lambda_3 \le ...$; a weak direct Mpemba effect is $\mid a_2(T_h)\mid < \mid a_2(T_c)\mid$ for some $T_h > T_c > T_b$, a strong one is $a_2(T_h) = 0$, and an inverse effect is $\mid a_2(T_c)\mid < \mid a_2(T_h)\mid$ for some $T_c < T_h < T_b$. Find necessary and sufficient conditions on $\{E_i, B_{ij}, T_b\}$ for the weak and strong, direct and inverse effects, for general n, stated as explicit inequalities or sign conditions (for example on the graph of barriers) that do not require computing the eigenvectors of the generator.

A theorem characterizing the parameter sets $\{E_i, B_{ij}, T_b\}$ for which $a_2(T_0)$ is nonmonotonic or vanishes, valid for all $n$.