Finite termination of min-sum auction I for unique maximum weight matchings

Let $\pi^*$ denote the maximum weight matching, and consider the min-sum auction I algorithm, whose step (4) includes the condition

$$\tilde{m}_{\alpha_{i_j}\rightarrow\beta_j}^k \geq \tilde{m}_{\beta_j\rightarrow\alpha_i}^{k-1}.$$

Finite-termination conjecture. If $\pi^*$ is unique, then the min-sum auction I algorithm terminates after finitely many iterations when this condition is removed from step (4).

The conjecture concerns whether uniqueness of the maximum weight matching suffices for finite termination of the modified algorithm. The source reports simulation evidence supporting it, but provides no proof or resolution.