Center-manifold conjecture for the modified curve-shortening flow

Consider the modified curve-shortening PDE

$$\partial_t f=\rho(x,f)\partial_{xx}f,$$

where $\rho$ is positive except at finitely many quadratic zeros along the $x$-axis. Let $y_i=|f(x_i)|/\pi$ for $i\in\mathbb Z/n$, where the $x_i$ are the punctures, and consider the associated $n$-dimensional ODE system

$$\frac{\dot{y}_i}{y_i}=\frac{\epsilon_{i-1}\epsilon_i}{m_i}y_{i-1}-\left(\frac{1}{m_i}+\frac{1}{m_{i+1}}\right)y_i+\frac{\epsilon_i\epsilon_{i+1}}{m_{i+1}}y_{i+1}.$$

Center-manifold conjecture. The PDE has an $n$-dimensional center manifold on which its flow is approximated by this ODE system, in the sense that the error terms of solutions are bounded in the coordinates $\log(y_i)$. The conjecture would make the heuristic finite-dimensional reduction precise and relate the flow asymptotics to the partially wrapped Fukaya category of the punctured cylinder; the supplied text does not state whether it is proved or remains open.