Classification conjecture for polar-degree-two hypersurfaces with isolated singularities

Let $V(h)\subseteq\mathbb{P}^n$ be a projective hypersurface defined by a homogeneous polynomial $h$ of degree $d$, with only isolated singular points, and let its polar degree be $2$. Polar-degree-two classification conjecture. After a linear change of homogeneous coordinates, $V(h)$ is one of the twelve hypersurfaces listed in the source: three normal cubic surfaces in $(n,d)=(3,3)$, one degree-five plane arrangement involving two smooth conics and their common tangent, seven degree-four plane curves, and two degree-three plane curves, with the displayed defining equations and singularity types in the source statement. The claim proposes a complete classification of the isolated-singularity hypersurfaces of polar degree two; the supplied text gives no resolution status.