Critical invariant-circle breakdown threshold conjecture for twist maps

Let $\rho$in$(0,1)$ be a Diophantine number, and let $F$ be a twist map of the form given in the source. Denote by $\Lambda(\rho)$ the parameter threshold at which the invariant circle with rotation number $\rho$ becomes critical. Critical invariant-circle threshold conjecture. There is a number $\Lambda(\rho)$ such that, when $|\lambda|>\Lambda(\rho)$, no invariant circle with rotation number $\rho$ exists, while, when $|\lambda|<\Lambda(\rho)$, an analytic invariant circle with rotation number $\rho$ exists; the invariant circle is critical when $|\lambda|=\Lambda(\rho)$. This expresses the expected sharp transition between analytic invariant circles and their breakdown as the coupling parameter increases; the source presents it as a widely held belief rather than a proved result.