Curvature flow convergence for Markovian weighted graphs

Let $(G,P_0)$ be an initial weighted graph and let $(G,P(t))$ be the curvature flow associated to it, with weighting scheme $P(t)$ at time $t$. A limit $P^ \infty$ is a weighting scheme satisfying $P^ \infty = \lim_{t \to \infty}P(t)$.

Curvature flow convergence conjecture. The curvature flow converges for any initial condition $(G,P_0)$, that is, $(G,P(t))$ has a well-defined limit

$$P^\infty = \lim_{t \to \infty} P(t).$$

The authors state that they do not know any initial condition for which the curvature flow fails to converge, motivating this conjecture. The preceding results show that whenever the flow converges, its limit is curvature sharp; convergence for arbitrary initial conditions remains open.