  We consider the mass spectrum of the 3$d$ 3-state Potts model in the broken
phase (a) near the second order Ising critical point in the temperature -
magnetic field plane and (b) near the weakly first order transition point at
zero magnetic field. In the case (a), we compare the mass spectrum with the
prediction from universality of mass ratios in the 3$d$ Ising class; in the
case (b), we determine a mass ratio to be compared with the corresponding one
in the spectrum of screening masses of the (3+1)$d$ SU(3) pure gauge theory at
finite temperature in the deconfined phase near the transition. The agreement
in the comparison in the case (a) would represent a non-trivial test of
validity of the conjecture of spectrum universality. A positive answer to the
comparison in the case (b) would suggest the possibility to extend this
conjecture to weakly first order phase transitions.
