The positive mass conjecture for asymptotically flat gravitational instantons

An AF gravitational instanton is an asymptotically flat gravitational instanton, denoted $(M,\mathbf{g})$. For toric AF instantons, the asymptotic expansion defines an invariant $m$ called the mass.

Positive mass conjecture. To any AF gravitational instanton $(M,\mathbf{g})$, one can assign a geometric invariant called the mass $m$, coinciding with the invariant $m$ for toric instantons, such that

$$m\geq 0,$$

with equality if and only if $(M,\mathbf{g})$ is flat.

The conjecture proposes an ADM-type mass for general AF gravitational instantons and extends the positive mass phenomenon known in related settings. The paper states it as a natural conjecture, but the supplied text gives no resolution.