Generic finite-graph conjecture for singular sets of two-dimensional Hamilton–Jacobi solutions

Let $B$ be the closed unit ball in $obreak\mathbb{R}^2$, and let $A(x)$ be a $C^\infty$ map from $B$ to the space of positive definite $2\times 2$ matrices. Let $u$ be the viscosity solution of

$$\begin{cases} \langle A(x)\nabla u(x),\nabla u(x)\rangle=1&\text{in }B,\\ u=0&\text{on }\partial B. \end{cases}$$

Generic finite-graph conjecture. For a generic such function $A(x)$, the closure of the singular set of $u$ is a finite graph.

In the analytic case, the corresponding finiteness result can be proved, whereas for arbitrary smooth metrics the closure of the singular set can be as wild as a non-triangulable cut locus. The conjecture is motivated by the fact that, for a generic smooth metric on a surface, cut loci are triangulable and have no points of order greater than three; the claimed generic result for viscosity solutions remains open in the source.