  We measure low-temperature micro-photoluminescence spectra along a MoS$_2$
nanotube, which exhibit the peaks of the optical whispering gallery modes below
the exciton resonance. The variation of the position and intensity of these
peaks is used to quantify the change of the nanotube geometry. The width of the
peaks is shown to be determined by the fluctuations of the nanotube wall
thickness and propagation of the detected optical modes along the nanotube. We
analyse the dependence of the energies of the optical modes on the wave vector
along the nanotube axis and demonstrate the potential of the high-quality
nanotubes for realization of the strong coupling between exciton and optical
modes with the Rabi splitting reaching 400 meV. We show how the formation of
exciton-polaritons in such structures will be manifested in the
micro-photoluminescence spectra.
