  Using cosmological particle hydrodynamical simulations and uniform
ultraviolet backgrounds, we compare Lyman-$\alpha$ forest flux spectra
predicted by the conventional cold dark matter (CDM) model, the free-particle
wave dark matter (FP$\psi$DM) model and extreme-axion wave dark matter
(EA$\psi$DM) models of different initial axion field angles against the BOSS
Lyman-$\alpha$ forest absorption spectra with a fixed boson mass $m_b\sim
10^{-22}$eV. We recover results reported previously (Ir\v{s}i\v{c} et al.
2017b; Armengaud et al. 2017) that the CDM model agrees better with the BOSS
data than the FP$\psi$DM model by a large margin, and we find the difference of
total $\chi^2$'s is $120$ for $420$ data bins. These previous results demand a
larger boson mass by a factor $>10$ to be consistent with the date and are in
tension with the favoured value determined from local satellite galaxies. We
however find that such tension is removed as some EA$\psi$DM models predict
Lyman-$\alpha$ flux spectra agreeing better with the BOSS data than the CDM
model, and the difference of total $\chi^2$'s can be as large as $24$ for the
same bin number. This finding arises with no surprise since EA$\psi$DM models
have unique spectral shapes with spectral bumps in excess of the CDM power near
the small-scale cutoff typical of $\psi$DM linear matter power spectra as well
as more extended cutoffs than FP$\psi$DM (Zhang & Chiueh 2017a,b).
