Non-degeneracy conjecture for Riemannian Kottler metrics

Let $P_L$ be the linearization, at a Riemannian Kottler metric with negative cosmological constant, of the $TT$-gauge-fixed Einstein operator. Here $K$ is the constant sectional curvature of the horizon, $\boldsymbol{\mu}$ is the mass parameter, and $L^2$ refers to the natural square-integrability condition for perturbations. Non-degeneracy conjecture. $P_L$ has no $L^2$-kernel except if $K=1$ and the mass parameter is

$$\mu=\mu_c:=\frac{n}{n+1}\left(\ell\sqrt{\frac{n-1}{n+1}}\right)^{n-1}.$$

The conjecture concerns the absence of infinitesimal Einstein deformations modulo the gauge fixing, and is motivated by the construction of stationary black hole spacetimes from non-degenerate Riemannian Kottler metrics. The paper proves the asserted non-degeneracy in dimension $n+2=4$ away from the critical spherical mass and for open ranges of mass parameters when $n>2$, but the full statement is not established here.