Uniqueness of tangent cones with immersed special Legendrian links

Under the assumption of integrability of the smooth Jacobi fields and a condition on the intersection pattern of the link, we prove uniqueness of special Lagrangian tangent cones in ${\mathbb{C}}^3$ whose links are compact immersed special Legendrian surfaces with transverse double points. The proof is based on a discrete Lojasiewicz type inequality, which comes from a gluing obstruction mechanism.