Uniqueness of some cylindrical tangent cones to special Lagrangians

We show that if an exact special Lagrangian $N\subset {\mathbb{C}}^n$ has a multiplicity one, cylindrical tangent cone of the form ${\mathbb{R}}^k\times \mathbf{C}$ where $\mathbf{C}$ is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $\mathbf{C}$ satisfies an integrability condition. This applies, for example, when $\mathbf{C}={\mathbf{C}}_{HL}^m$ is the Harvey-Lawson $T^{m-1}$ cone for $m\mathrm{\ne }8,9$.