Dafermos–Holzegel static uniqueness conjecture for the Eguchi-Hanson mass gap

Let $a>0$, let $n$ be the integer determining the topology $S^3/\mathbb{Z}_n$, and consider static, globally regular, asymptotically locally anti-de Sitter solutions of the five-dimensional Einstein vacuum equations with that topology. Let $M$ be the mass, and let $M_{\mathrm{EH},a}$ be the Eguchi-Hanson-AdS mass. The mass-gap condition is

$$M_{\mathrm{EH},a}<M<0.$$

Dafermos–Holzegel conjecture. There are no static, globally regular asymptotically locally AdS solutions to the five-dimensional Einstein vacuum equations with topology $S^3/\mathbb{Z}_n$ and mass $M$ satisfying

$$M_{\mathrm{EH},a}<M<0.$$

The conjecture is motivated by static uniqueness for exact anti-de Sitter space. The supplied text does not report a proof or refutation.