Local Debye-potential generation conjecture for linearized gravity on vacuum type II spacetimes

Let the background be a vacuum type II spacetime, and consider real solutions of the linearized Einstein vacuum equations on it. A metric generated by a Debye potential is a metric perturbation obtained from a Debye potential through the construction considered in the paper.

Debye-potential generation conjecture. All real solutions of the linearized Einstein vacuum equations on a vacuum type II background can be locally obtained, up to gauge, as the real part of the metric generated by a Debye potential.

For vacuum type D spacetimes, an analogous assertion is sometimes assumed, and recent progress is known in the Schwarzschild and Kerr cases. The conjecture is motivated by the type II result that the linearized curvature of the complex metric perturbation generated by a Debye potential has a simple, half-type N structure; its status for general vacuum type II backgrounds is not established.