Equivariant Dugundji characterization of metric G-ANE spaces

Let $G$ be a compact group. A metric $G$-ANE space is a metric $G$-space $\Bbb X\in G$-$\operatorname{ANE}$. An arbitrarily fine domination of $\Bbb X$ by $G$-$\operatorname{CW}$-complexes means that, for every cover $\omega\in\operatorname{cov}\Bbb X$, there are a $G$-$\operatorname{CW}$-complex $\Bbb Y$ and $G$-maps

$$\Bbb X\xrightarrow{f}\Bbb Y\xrightarrow{g}\Bbb X$$

such that $g\circ f$ and $\operatorname{Id}_{\Bbb X}$ can be joined by an $\omega$-$G$-homotopy. Equivariant Dugundji characterization. Every metric $G$-space $\Bbb X\in G$-$\operatorname{ANE}$ admits an arbitrarily fine domination by $G$-$\operatorname{CW}$-complexes: for each cover $\omega\in\operatorname{cov}\Bbb X$, there exist a $G$-$\operatorname{CW}$-complex $\Bbb Y$ and $G$-maps $\Bbb X\xrightarrow{f}\Bbb Y\xrightarrow{g}\Bbb X$ such that $g\circ f$ and $\operatorname{Id}_{\Bbb X}$ can be joined by an $\omega$-$G$-homotopy. This is presented as the equivariant analogue of Dugundji's characterization of absolute neighborhood retracts by arbitrarily fine domination by simplicial complexes. The supplied text does not indicate whether the statement has been proved or remains open.