Higher-order gauge-transformation conjecture for metric perturbations

Fix an order $n$ and let $J_i$, ${}_nJ_0^+$, ${\cal C}_{n-1}$, $\xi_{(i)}$, and ${}_{\cal X}^{(i)}X$, ${}_{\cal Y}^{(i)}X$ denote the combinatorial sets, coefficients, gauge-transformation generators, and gauge vectors used in the displayed definition of ${}^{(n)}\hat H_{ab}$ and its transformation law. The expression formed by the two sums below is a differential operator acting on tensor fields.

Higher-order gauge-transformation conjecture. There exists a vector field $\hat\sigma_{(n)}^a$ such that

$$\begin{aligned} &n!\sum_{\\{j_l\\}\in J_n\backslash{}_nJ_0^+} {\cal C}_{n-1}(\\{j_l\\})\left(\pounds_{\xi_{(1)}}^{j_1}\cdots\pounds_{\xi_{(n-1)}}^{j_{n-1}}+\pounds_{-{}_{\cal Y}^{(1)}X}^{j_1}\cdots\pounds_{-{}_{\cal Y}^{(n-1)}X}^{j_{n-1}}-\pounds_{-{}_{\cal X}^{(1)}X}^{j_1}\cdots\pounds_{-{}_{\cal X}^{(n-1)}X}^{j_{n-1}}\right)\\\\ &+n!\sum_{i=1}^{n-1}\sum_{\\{j_l\\}\in J_i}{\cal C}_{n-1}(\\{j_l\\})\pounds_{-{}_{\cal Y}^{(1)}X}^{j_1}\cdots\pounds_{-{}_{\cal Y}^{(n-1)}X}^{j_{n-1}}\sum_{\\{k_m\\}\in J_{n-i}}{\cal C}_{n-1}(\\{k_m\\})\pounds_{\xi_{(1)}}^{k_1}\cdots\pounds_{\xi_{(n-1)}}^{k_{n-1}}\\\\ &=\pounds_{\hat\sigma_{(n)}}. \end{aligned}$$

The conjecture is intended to show that the remaining higher-order gauge transformation is itself a Lie derivative, enabling the construction of gauge-invariant variables at order $n$. The supplied text gives no resolution or further context establishing whether this claim is proved.