Boundary characterization at regular isolated-set points

Let $\mathcal{S}_{nt}^{iso}(\mu)$ denote the isolated-point part of the non-tangential singular set, let $\mathcal{E}$ be the relevant edge set, and let $\partial\mathcal{L}(x)$ denote the boundary fiber over $x$. Assume that $x\in\partial\mathcal{S}_{nt}^{iso}(\mu)$ and that $x$ is a regular point. Boundary characterization conjecture. Then

$$(x,1)\in\overline{\mathcal{E}}\quad\text{and}\quad\partial\mathcal{L}(x)=\\{(x,1)\\}.$$

The surrounding argument proves the corresponding one-sided boundary convergence under additional density assumptions. The conjecture asserts the full boundary description at every regular boundary point of the isolated singular set.