Instability conjecture for cosmological horizons in static solutions

Let $(M,g_0,u)$ be a static solution to problem~, and call a horizon of cosmological type when it has the cosmological-horizon character described in the paper. A horizon is unstable when its associated stability operator has a negative direction.

Cosmological-horizon instability conjecture. Every horizon of cosmological type is necessarily unstable. In particular, every static solution to problem~ has at most one horizon of cosmological type.

The conjecture is motivated by the distinction between cosmological and black-hole horizons and would remove the connected-cosmological-horizon hypothesis from the three-dimensional Black Hole Uniqueness Theorem. Its status is not resolved in the supplied text.