Counterexamples to the conjectured ordering between the waiting-time bound and the thermodynamic uncertainty bound on entropy production

Two widely used model-free lower bounds on the steady-state entropy production rate of a continuous-time Markov jump process are the thermodynamic uncertainty relation (TUR) bound $\sigma_\text{TUR}$, derived from the mean and variance of a current, and the waiting-time distribution (WTD) bound $\sigma_\mathcal{L}$, derived from the irreversibility of partially observed transition sequences together with their waiting times. It has been conjectured that $\sigma_{\mathcal L}$ is always at least as tight as $\sigma_{\mathrm{TUR}}$ when both are constructed from the same partially observed link. Here we provide four-state counterexamples in a nonequilibrium steady state where $\sigma_{\mathcal L}<\sigma_{\mathrm{TUR}}$. This result shows that no universal ordering exists between these two inference bounds under partial observation.