Dimca–Papadima conjecture on homaloidal hypersurfaces with isolated singularities

Let $V(h)\subseteq\mathbb{P}^n$ be a homaloidal hypersurface defined by a homogeneous polynomial $h$ of degree $d\geq 3$, and suppose that $V(h)$ has only isolated singular points. Dimca–Papadima conjecture. There are no such homaloidal hypersurfaces when $n\geq 3$. The conjecture concerns the incompatibility between birationality of the gradient map and isolated singularities in dimensions at least three; the supplied text does not indicate whether it has been resolved.