Infinitely many folds on the upper solution branch

Let $\varepsilon>0$ be fixed and sufficiently small, and consider the upper solution branch of the bifurcation diagram for the arclength system. As $|z(0)|\to1^-$, let $\lambda>0$ denote the corresponding parameter.

Upper-branch fold conjecture. The upper solution branch undergoes infinitely many folds, and $\lambda$ tends to a finite value bounded away from zero.

This conjecture predicts the limiting structure of the singular solution branch for the prescribed mean curvature problem with singular nonlinearity. It is motivated by the asymptotic analysis and numerical continuation described in the source; its resolution is not established here.