The extended-spectrum characterization of interweaving relations for finite Markov generators

Let $L$ and $\widetilde L$ be irreducible Markov generators on finite state spaces $V$ and $\widetilde V$. The extended spectrum of a generator consists of its eigenvalues together with the dimensions of the associated Jordan blocks, where inclusion permits smaller or equal block dimensions.

Extended-spectrum interweaving conjecture. There exists an interweaving relation from $(\exp(tL))_{t\geq 0}$ to $(\exp(t\widetilde L))_{t\geq 0}$ if and only if the extended spectrum of $L$ is included in that of $\widetilde L$.

This conjecture proposes a spectral characterization of interweaving relations in the finite irreducible setting. The source presents it as a conjecture for future investigation, and no resolution is supplied.