Structure of non-degenerate subgraphs of curvature-flow limits

Let $(G,P_0)$ be a non-degenerate unmixed Markovian weighted graph whose normalized curvature flow converges to a not totally degenerate limit $(G,P^\infty)$. Let $W\subset V$ be the vertices incident to non-degenerate edges of the limit, let $G_W$ be the subgraph consisting of $W$ and all non-degenerate edges of $(G,P^\infty)$, and let $P_W$ be the restriction of $P^\infty$ to $W$.

Non-degenerate-subgraph conjecture. The subgraph $G_W$ coincides with the subgraph of $G$ induced by $W$, all transition rates from $W$ to $V\setminus W$ are zero, and $(G_W,P_W)$ is a non-degenerate Markovian weighted graph that is itself curvature sharp.

This conjecture describes the structure of a non-totally-degenerate flow limit and its surviving transition rates; its resolution is not given here.