Nonexistence of globally hyperbolic developments for quasilinear wave equations

Consider a quasilinear wave equation of the form

$$, together with initial data. A **unique globally hyperbolic development** (UGHD) is a globally hyperbolic development that is unique among globally hyperbolic developments of the same initial data. **Nonexistence conjecture.** There are quasilinear wave equations of the form$$

for which there exists initial data such that there does not exist any UGHD.

This is presented as an analogue that is expected to fail for more general quasilinear wave equations, namely equations that are neither subluminal nor superluminal. The source does not provide a resolution.