Lipschitz structure conjecture for branch points of minimizers

Let $u$ be a minimizer of the functional in on some open set $\Omega\subset\mathbb R^n$, and let $\Gamma_{\mathrm{BP}}(u)$ denote its branch-point set. For a relatively compact set $D\Subset\Omega$, consider $D\cap\Gamma_{\mathrm{BP}}(u)$. Branch-point structure conjecture. The set $D\cap\Gamma_{\mathrm{BP}}(u)$ is locally contained in finitely many Lipschitz $(n-2)$-dimensional submanifolds. This conjecture proposes an analogue of the local structure known for singular sets of area-minimizing surfaces; the source gives no resolution, so the assertion remains open.