White's discreteness conjecture for analytic boundaries in dimension two

Let $T$, $\Sigma$, and $\Gamma$ be as in the main assumptions, with $m=2$, and suppose that $\Sigma$ and $\Gamma$ are real analytic. The boundary and interior singular sets are the singular sets associated with the mass-minimizing current $T$.

White's discreteness conjecture. The union of the boundary and interior singular sets is discrete.

The source identifies this as the nonlinear counterpart of the linearized analytic-interface conjecture and says that it is widely open.