Further clarification on quasinormal modes/circular null geodesics correspondence

The well-known correspondence between quasinormal modes of any stationary, spherically symmetric and asymptotically flat or de Sitter black hole and parameters of the circular null geodesic was initially claimed for gravitational and test field perturbations. According to this correspondence the real and imaginary parts of the ℓ≫n quasinormal mode (where ℓ and n are multipole and overtone numbers respectively) are multiples of the frequency and instability timescale of the circular null geodesics respectively. Later it was shown that the correspondence is guaranteed only for test fields and may be broken for gravitational and other non-minimally coupled fields. Here, we further specify the correspondence and prove that even when it is guaranteed, it may not represent the full spectrum of the ℓ≫n modes, missing the quasinormal frequencies which cannot be found by the standard WKB method. In particular we show that this always happens for an arbitrary asymptotically de Sitter black holes and further argue that, in general, this might be related to sensitivity of the quasinormal spectrum to geometry deformations near the boundaries.