Conjectured structure of the top stratum in the no-sign obstacle problem

Let $u$ be a solution of the no-sign obstacle problem

$$\Delta u=\chi_{\{u\ne 0\}}\quad\text{in }B_1,$$

and let $\Sigma_{n-1}$ denote the top stratum of the free boundary, with $\Sigma_{n-1,a}$ its anomalous part. Here $\dim_{\mathcal H}$ denotes Hausdorff dimension. Top-stratum structure conjecture.

(a) For $n=2$, $\Sigma_1$ is locally contained in a $C^{1,1/2}$ curve.

(b) For $n\ge 3$, $\Sigma_{n-1,a}$ is a discrete set if $n=3$, while

$$\dim_{\mathcal H}(\Sigma_{n-1,a})\le n-3$$

when $n\ge 4$.

(c) For $n\ge 3$, $\Sigma_{n-1}$ is locally covered by a $C^{1,\alpha_0}$ $(n-1)$-dimensional manifold for some $\alpha_0=\alpha_0(n)\in(0,1)$.

The conjecture proposes a precise geometric description of the highest-dimensional free-boundary stratum, including improved regularity in dimension two, a dimension bound for anomalous points, and local manifold structure in higher dimensions. The surrounding discussion presents these assertions as conjectural; no resolution is supplied in the source.