  The multi-Regge limit of scattering amplitudes in strongly-coupled
$\mathcal{N}=4$ super Yang-Mills is described by the large mass limit of a set
of thermodynamic Bethe ansatz (TBA) equations. A non-trivial remainder function
arises in this setup in certain kinematical regions due to excitations of the
TBA equations which appear during the analytic continuation into these
kinematical regions. So far, these analytic continuations were carried out on a
case-by-case basis for the six- and seven-gluon remainder function. In this
note, we show that the set of possible excitations appearing in any analytic
continuation in the multi-Regge limit for any number of particles is rather
constrained. In particular, we show that the BFKL eigenvalue of any possible
Reggeon bound state is a multiple of the two-Reggeon BFKL eigenvalue appearing
in the six-gluon case.
