  We present a $^{31}$P nuclear magnetic resonance (NMR) investigation of
BaCdVO(PO$_4$)$_2$ focusing on the nearly saturated regime between
$\mu_0H_{c1}$ = 4.05 T and $\mu_0H_{c2}$ = 6.5 T, which used to be considered a
promising candidate for a spin-nematic phase. NMR spectra establish the absence
of any dipolar order there, whereas the weak field dependence of the
magnetization above $H_{c1}$ is accounted for by Dzyaloshinskii-Moriya
interaction terms. The low-energy spin dynamics (fluctuations), measured by the
nuclear spin-lattice relaxation rate $T_1^{-1}$, confirms the continuity of
this phase and the absence of any low-temperature phase transition.
Unexpectedly, the spin dynamics above $H_{c1}$ is largely dominated by
two-magnon processes, which is expected above the saturation field of a
spin-nematic phase, but not inside. This shows that BaCdVO(PO$_4$)$_2$ is
indeed close to a spin-nematic instability; however, this phase is not
stabilized. We thus confirm recent theoretical predictions that the
spin-nematic phase can be stabilized, at most, in an extremely narrow field
range close to saturation or is rather narrowly avoided [Jiang et al., Phys.
Rev. Lett. 130, 116701 (2023)].
