  Quantum magnetic phases near the magnetic saturation of triangular-lattice
antiferromagnets with XXZ anisotropy have been attracting renewed interest
since it has been suggested that a nontrivial coplanar phase, called the
$\pi$-coplanar or $\Psi$ phase, could be stabilized by quantum effects in a
certain range of anisotropy parameter $J/J_z$ besides the well-known 0-coplanar
(known also as $V$) and umbrella phases. Recently, Sellmann $et$ $al$. [Phys.
Rev. B {\bf 91}, 081104(R) (2015)] claimed that the $\pi$-coplanar phase is
absent for $S=1/2$ from an exact-diagonalization analysis in the sector of the
Hilbert space with only three down-spins (three magnons). We first reconsider
and improve this analysis by taking into account several low-lying eigenvalues
and the associated eigenstates as a function of $J/J_z$ and by sensibly
increasing the system sizes (up to 1296 spins). A careful identification
analysis shows that the lowest eigenstate is a chirally antisymmetric
combination of finite-size umbrella states for $J/J_z\gtrsim 2.218$ while it
corresponds to a coplanar phase for $J/J_z\lesssim 2.218$. However, we
demonstrate that the distinction between 0-coplanar and $\pi$-coplanar phases
in the latter region is fundamentally impossible from the symmetry-preserving
finite-size calculations with fixed magnon number.} Therefore, we also perform
a cluster mean-field plus scaling analysis for small spins $S\leq 3/2$. The
obtained results, together with the previous large-$S$ analysis, indicate that
the $\pi$-coplanar phase exists for any $S$ except for the classical limit
($S\rightarrow \infty$) and the existence range in $J/J_z$ is largest in the
most quantum case of $S=1/2$.
