We report a V51 nuclear magnetic resonance investigation of the frustrated spin-1/2 chain compound LiCuVO4, performed in pulsed magnetic fields and focused on high-field phases up to 56 T. For the crystal orientations H¥c and H¥b, we find a narrow field region just below the magnetic saturation where the local magnetization remains uniform and homogeneous, while its value is field dependent. This behavior is the first microscopic signature of the spin-nematic state, breaking spin-rotation symmetry without generating any transverse dipolar order, and is consistent with theoretical predictions for the LiCuVO4 compound.