Necessary local normal form conjecture for conformally flat transverse Riemann-Lorentz manifolds

Let $(M,\mathcal{C})$ be a transverse Riemann-Lorentz conformal manifold of dimension $m\geq 4$, with singular set $\Sigma$. Around each singular point $p\in\Sigma$, consider a coordinate system $(\mathbb{U},x)$ and a metric $g\in\mathcal{C}$, where $\tau=0$ is a local equation for $\Sigma$. Necessary local normal form conjecture. If the Weyl curvature satisfies $W=0$, then there exist such $(\mathbb{U},x)$ and $g$ for which

$$g=\sum_{i=0}^{m-1}(dx^i)^2+\tau(dx^m)^2.$$

This asserts a necessary local normal form for conformally flat transverse Riemann-Lorentz structures near their singular set. The source presents it as a conjecture; no resolution is supplied in the given text.