Global convergence conjecture for fractional combinatorial Calabi flow with surgery

Let $(M,V,\varepsilon)$ be a marked weighted connected closed surface with $\varepsilon:V\rightarrow\{0,1\}$, and suppose there exists a PL or PH metric generated by a discrete conformal structure with combinatorial curvature $\overline{K}$. For any $s\in\mathbb{R}$ and any initial PL or PH metric on $(M,V,\varepsilon)$ generated by a discrete conformal structure, Global convergence conjecture. the solution of fractional combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after a finite number of surgeries. This extends the known convergence results for the cases $s=0$ and $s=1$ to arbitrary real $s$; because the fractional combinatorial Calabi flow is generally not a gradient flow, finiteness of surgeries is not known in general.