  It has been suggested that the internal dynamics of dwarf spheroidal galaxies
(dSphs) can be used to test whether or not ultralight axions with $m_a\sim
10^{-22}\text{eV}$ are a preferred dark matter candidate. However, comparisons
to theoretical predictions tend to be inconclusive for the simple reason that
while most cosmological models consider only dark matter, one observes only
baryons. Here we use realistic kinematic mock data catalogs of Milky Way dSph's
to show that the "mass-anisotropy degeneracy" in the Jeans equations leads to
biased bounds on the axion mass in galaxies with unknown dark matter halo
profiles. In galaxies with multiple chemodynamical components this bias can be
partly removed by modelling the mass enclosed within each subpopulation.
However, analysis of the mock data reveals that the least-biased constraints on
the axion mass result from fitting the luminosity-averaged velocity dispersion
of the individual chemodynamical components directly. Applying our analysis to
two dSph's with reported stellar subcomponents, Fornax and Sculptor, and
assuming that the halo profile has not been acted on by baryons, yields core
radii $r_{c}>1.5$ kpc and $r_c> 1.2$ kpc respectively, and $m_a<0.4\times
10^{-22}\text{eV}$ at 97.5\% confidence. These bounds are in tension with the
number of observed satellites derived from simple (but conservative) estimates
of the subhalo mass function in Milky Way-like galaxies. We discuss how
baryonic feedback might affect our results, and the impact of such a small
axion mass on the growth of structures in the Universe.
