Global persistence conjecture for simple eigenvector solutions

Let $(x_*,0,\lambda_*)$ be a simple solution of the eigenvalue problem, meaning that the first three hypotheses stated in the paper hold. Suppose that $G$ and $H$ are separable, and that $N$ and $C$ are compact. Global persistence conjecture. The set of nontrivial solutions has a connected subset whose closure contains $(x_*,0,\lambda_*)$ and is either unbounded or meets a trivial solution $(x^*,0,\lambda^*)$ with $\lambda^*\ne\lambda_*$. The conjecture would sharpen the preceding global continuation theorem from eigenpairs to eigenvector solutions in the infinite-dimensional setting; the authors state that they have not been able either to prove or to disprove it.