Gauge-invariant decomposition conjecture for metric perturbations

Let $g_{ab}$ be the background metric, and let $h_{ab}$ be a symmetric rank-two tensor field whose gauge transformation between gauge choices ${\cal X}$ and ${\cal Y}$, generated by the vector field $\xi$, is

$${}_{\\;{\cal Y}}\\!h_{ab}-{}_{\\;{\cal X}}\\!h_{ab}=\pounds_{\xi}g_{ab}.$$

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

$$h_{ab}={\cal H}_{ab}+\pounds_Xg_{ab},$$

and, under the gauge transformation, they satisfy

$${}_{\\;{\cal Y}}\\!{\cal H}_{ab}-{}_{\\;{\cal X}}\\!{\cal H}_{ab}=0,\qquad {}_{\\;{\cal Y}}X^a-{}_{\\;{\cal X}}X^a=\xi^a.$$

This decomposition separates the gauge-invariant part of a metric perturbation from its gauge-variant Lie-derivative part and is the starting point for constructing gauge-invariant variables at first and second order. The supplied source presents it as a conjectural starting point and gives no resolution evidence, so its status remains open.