The critical-length conjecture for standard components of linear slices

Let ${\mathcal L}_c$ denote the linear slice at parameter $c$, and distinguish its standard component from its non-standard components. Critical-length conjecture. There exists a unique $c_0$ such that ${\mathcal L}_c$ coincides with the standard component for any $c \leq c_0$, while ${\mathcal L}_c$ contains infinitely many non-standard components for any $c>c_0$. This conjecture is supported by numerical experiments and describes a transition in the topology of the linear slices at the critical value $c_0$.