Nonexistence conjecture for isolated cylindrical singularities with stable connected-link cones

Let $Y$ be an exact special Lagrangian submanifold in $\mathbf C^{n+1}$ with tangent cone $C\times\mathbf R$ at $0$. Assume that $C$ is stable, meaning that the only linear and quadratic harmonic functions on $C$ correspond to translations and rotations, and that $C$ has a smooth connected link. Nonexistence conjecture. Then $0$ is not an isolated singularity of $Y$. This conjecture predicts that stable cones with smooth connected links cannot occur as cylindrical tangent cones at isolated singularities of exact special Lagrangian submanifolds; the source presents it as a proposed consequence of the connectedness obstruction and does not state a resolution.