  The pattern of a large approximate degeneracy of the excited hadron spectra
(larger than the chiral restoration degeneracy) is present in the recent
experimental report of Bugg. Here we try to model this degeneracy with state of
the art quark models. We review how the Coulomb Gauge chiral invariant and
confining Bethe-Salpeter equation simplifies in the case of very excited
quark-antiquark mesons, including angular or radial excitations, to a Salpeter
equation with an ultrarelativistic kinetic energy with the spin-independent
part of the potential. The resulting meson spectrum is solved, and the excited
chiral restoration is recovered, for all mesons with J > 0. Applying the
ultrarelativistic simplification to a linear equal-time potential, linear Regge
trajectories are obtained, for both angular and radial xcitations. The spectrum
is also compared with the semi-classical Bohr-Sommerfeld quantization relation.
However the excited angular and radial spectra do not coincide exactly. We then
search, with the classical Bertrand theorem, for central potentials producing
always classical closed orbits with the ultrarelativistic kinetic energy. We
find that no such potential exists, and this implies that no exact larger
degeneracy can be obtained in our equal-time framework, with a single principal
quantum number comparable to the non-relativistic Coulomb or harmonic
oscillator potentials. Nevertheless we find plausible that the large
experimental approximate degeneracy will be modelled in the future by quark
models beyond the present state of the art.
