True-solution conjecture for gluing small black holes along timelike geodesics

Let $V\subset M$ be a precompact open set. For sufficiently small $\epsilon>0$, let $g_\epsilon$ be the formal solution on $M_\epsilon$ obtained by gluing small Kerr black holes along the prescribed timelike geodesic configuration, and let $\mathcal U^\circ$ denote the relevant interior region. A tensor $h_\epsilon$ is required to be smooth on $\overline{V}\cap M_\epsilon$, vanish to infinite order at $\epsilon=0$, and vanish outside the domain of influence of a compact subset of $\mathcal U^\circ$. True solution conjecture. There exists $\epsilon_0>0$ such that, for every $0<\epsilon<\epsilon_0$, one can choose such an $h_\epsilon$ for which

$$\operatorname{Ric}(g_\epsilon+h_\epsilon)-\Lambda(g_\epsilon+h_\epsilon)=0.$$

The conjecture asks whether the formal gluing construction can be corrected to an actual solution of the Einstein vacuum equations. Such a correction would be needed, for example, to apply nonlinear stability results to extreme mass-ratio mergers; the supplied source does not state that this has been proved or disproved.