Nakamura's gauge-invariant decomposition conjecture for linear metric perturbations

Let ${{\mathscr M}}_{\epsilon}$ be a physical spacetime, let ${{\mathscr M}}$ be its background spacetime, and let $h_{ab}$ be a perturbative metric tensor pulled back from ${{\mathscr M}}_{\epsilon}$ to ${{\mathscr M}}$. For gauge choices ${{\mathscr X}}_{\epsilon}$ and ${{\mathscr Y}}_{\epsilon}$, suppose the gauge transformation is

$${}_{{{\mathscr Y}}}\!h_{ab}-{}_{{{\mathscr X}}}\!h_{ab}=\pounds_{\xi_{(1)}}g_{ab},$$

where $g_{ab}$ is the background metric. Nakamura's gauge-invariant decomposition conjecture. There exist a tensor field ${{\mathscr F}}_{ab}$ and a vector field $Y^a$ such that

$$h_{ab}:={{\mathscr F}}_{ab}+\pounds_Yg_{ab},$$

with transformation rules

$${}_{{{\mathscr Y}}}\!{{\mathscr F}}_{ab}-{}_{{{\mathscr X}}}\!{{\mathscr F}}_{ab}=0, \qquad {}_{{{\mathscr Y}}}\!Y^a-{}_{{{\mathscr X}}}\!Y^a=\xi^a_{(1)}.$$

This conjecture is the premise underlying Nakamura's higher-order gauge-invariant perturbation theory: ${{\mathscr F}}_{ab}$ is gauge invariant, while $Y^a$ carries the gauge dependence. The supplied text does not state whether the conjecture has been proved or remains open.