  We study Ricci-flat perturbations of gravitational instantons of Petrov type
D. Analogously to the Lorentzian case, the Weyl curvature scalars of extreme
spin-weight satisfy a Riemannian version of the separable Teukolsky equation.
As a step towards rigidity of the type D Kerr and Taub-bolt families of
instantons, we prove mode stability, i.e., that the Teukolsky equation admits
no solutions compatible with regularity and asymptotic (local) flatness.
