Bryant–Griffiths–Yang conjecture on the characteristic variety of the isometric embedding system

Let $g$ be a smooth metric near the origin in $\mathbb R^n$. For parameters ${\bf c}=\{c_i^{kj}\}_{k\ne j}$ satisfying $c_i^{kj}=c_i^{jk}$ for $j\ne k$, $c_i^{ii}=1$, and $c_i^{jj}=0$ for $i\ne j$, define matrices $A^1,\ldots,A^n$ by

$$(A^k)_{ij}=c_i^{kj}.$$

For $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb R^n$, let

$$P(\xi,{\bf c})=\sum_{k=1}^n\xi_kA^k$$

and define the characteristic variety by

$$\Sigma({\bf c})=\\{\xi\in\mathbb R^n\setminus\\{0\\}:\det P(\xi,{\bf c})=0\\}.$$

Bryant–Griffiths–Yang conjecture. The characteristic variety $\Sigma({\bf c})$ is not smooth in $\mathbb R^n\setminus\\{0\\}$ for every $n\ge 5$.

The characteristic variety governs the type and analytic behavior of the reduced linearized isometric embedding system. It is known to be generally smooth for $n=4$ and nonsmooth for $n=6,10,14,\ldots$, while the assertion for all dimensions $n\ge5$ remains open in the supplied source.