Uniqueness of tangent cones at analytic-boundary points

Let $T$, $\Sigma$, and $\Gamma$ be as in White's discreteness conjecture: $T$ is an area-minimizing $2$-dimensional integral current in dimension $m=2$, and $\Sigma$ and $\Gamma$ are real analytic. Let $p\in\Gamma$. A tangent cone to $T$ at $p$ is a limit of blow-ups of $T$ centered at $p$.

Tangent-cone uniqueness conjecture. There is a unique tangent cone to $T$ at $p$.

The source presents this as an approachable subproblem that, to the authors' knowledge, had not been addressed in the literature; its status is open.