Finite-time attainment of the stationary regime

Prove or disprove that the complete downward closed graph is the only connected graph for which the liquid bin model's stationary regime is reached in finite time.

Background

The paper proves exponential convergence of car trajectories, and hence of the liquid bin dynamics, toward a unique stationary trajectory. It observes that finite-time attainment occurs when the stationary downward closed graph is complete, because after a suitable time every newly moving car follows the stationary trajectory exactly. The authors conjecture that no other connected graph has this finite-time stabilization property.

References

In some cases, the stationary regime is reached in finite time. For instance, this is the case when the DC graph of the stationary configuration is complete. In this case, there is a time at which there is no car between road signs 1 and N. Then, one may check that any car starting to move after this time has the stationary trajectory. We conjecture that the complete graph is the only connected graph for which the stationary regime is reached in finite time.

— Wall-crossing phenomenon for the liquid bin model  (2504.00301 - Ramassamy et al., 1 Apr 2025) in Remark 3.10 in Section 3, Subsection 3.2

We conjecture that the complete graph is the only connected graph for which the stationary regime is reached in finite time.

— Wall-crossing phenomenon for the liquid bin model  (2504.00301 - Ramassamy et al., 1 Apr 2025) in Remark 3.??, Remark “finitetime” in Section 3, at the end of the stationary car model section