Monodromy of general hypersurfaces

Let $X$ be a general complex projective hypersurface in $\mathbb{P}^{n+1}$ of degree $d>1$. A point $P$ not in $X$ is called uniform if the monodromy group of the projection of $X$ from $P$ is isomorphic to the symmetric group. We prove that all the points in $\mathbb{P}^{n+1}$ are uniform for $X$, generalizing a result of Cukierman on general plane curves.