Decomposition conjecture for linear metric perturbations

Let $h_{ab}$ be a second-rank tensor field whose gauge transformation between gauge choices ${\cal X}_{\lambda}$ and ${\cal Y}_{\lambda}$ satisfies

$${}_{\cal Y}h_{ab}-{}_{\cal X}h_{ab}={\pounds}_{\xi_{(1)}}g_{ab}.$$

Decomposition conjecture. There exist a tensor field ${\cal H}_{ab}$ and a vector field $X^a$ such that

$$h_{ab}={\cal H}_{ab}+{\pounds}_{X}g_{ab},$$

with transformation laws

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

This conjecture is the starting point for constructing gauge-invariant variables in linear-order metric perturbation theory: ${\cal H}_{ab}$ is gauge invariant, while $X^a$ captures the gauge-variant part of the perturbation. The supplied source does not state whether the conjecture has been proved or disproved.