Equality characterization for the Riemannian ZAS inequality

Let $(M,g)$ be as in Theorem of the Riemannian ZAS inequality, with boundary $\Sigma$ consisting of zero area singularities, ADM mass $m$, and ZAS mass $m_{\operatorname{ZAS}}(\Sigma)$. Assume that $m=m_{\operatorname{ZAS}}(\Sigma)$.

Equality characterization conjecture. Then $(M,g)$ is isometric to a Schwarzschild metric with finitely many points deleted when $-\infty<m<0$, or to the flat metric on $\mathbb{R}^3$ with finitely many points deleted when $m=0$.

The conjecture asserts that deleted points are the only obstruction to uniqueness of the Schwarzschild ZAS in the equality case. The source presents it as an open question for non-harmonically regular ZAS.