  We theoretically study the superfluid phase of a strongly correlated
$^{173}$Yb Fermi gas near its orbital Feshbach resonance, by developing a
quantitative pair-fluctuation theory within a two-band model. We examine the
density excitation spectrum of the system and determine a stability phase
diagram. We find that the $^{173}$Yb Fermi gas is intrinsically metastable and
has a peculiar equation of state, due to the small but positive singlet
scattering length. The massive Leggett mode, arising from the fluctuation of
the relative phase of two order parameters, is severely damped. We discuss the
parameter space where an undamped Leggett mode may exist.
