A sharp Bernstein-type theorem for entire minimal graphs

We consider entire solutions $u$ to the minimal surface equation in $R^N$, with $N\ge 8,$ and we prove the following sharp result : if $N-7$ partial derivatives $\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_j}$ are bounded on one side (not necessarily the same), then $u$ is necessarily an affine function.