Huh's polar-degree-two classification conjecture for projective hypersurfaces

Let $V\subset \mathbb{P}^{n}$ be a projective hypersurface with only isolated singularities, defined by a homogeneous polynomial $f$ of degree $d$, and let the polar degree of $V$ be $2$. After a linear change of homogeneous coordinates, the listed possibilities are the following: a normal cubic surface containing a single line, with $n=3$, $d=3$, and $f=x_{0}x_{1}^{2}+x_{1}x_{2}^{2}+x_{1}x_{3}^{2}+x_{2}^{3}$; a normal cubic surface containing two lines, with $n=3$, $d=3$, and $f=x_{0}x_{1}x_{2}+x_{0}x_{3}^{2}+x_{1}^{3}$; a normal cubic surface containing three lines and three binodes, with $n=3$, $d=3$, and $f=x_{0}x_{1}x_{2}+x_{3}^{3}$; two smooth conics meeting at a single point and their common tangent, with $n=2$, $d=5$, and $f=x_{0}(x_{1}^{2}+x_{0}x_{2})(x_{1}^{2}+x_{0}x_{2}+x_{0}^{2})$; two smooth conics meeting at a single point, with $n=2$, $d=4$, and $f=(x_{1}^{2}+x_{0}x_{2})(x_{1}^{2}+x_{0}x_{2}+x_{0}^{2})$; a smooth conic, a tangent, and a line passing through the tangency point, with $n=2$, $d=4$, and $f=x_{0}(x_{0}+x_{1})(x_{1}^{2}+x_{0}x_{2})$; a smooth conic and two tangent lines, with $n=2$, $d=4$, and $f=x_{0}x_{2}(x_{1}^{2}+x_{0}x_{2})$; three concurrent lines and a line not meeting the center point, with $n=2$, $d=4$, and $f=x_{0}x_{1}x_{2}(x_{0}+x_{1})$; a cuspidal cubic and its tangent at the cusp, with $n=2$, $d=4$, and $f=x_{0}(x_{1}^{3}+x_{0}^{2}x_{2})$; a cuspidal cubic and its tangent at the smooth flex point, with $n=2$, $d=4$, and $f=x_{2}(x_{1}^{3}+x_{0}^{2}x_{2})$; a cuspidal cubic, with $n=2$, $d=3$, and $f=x_{1}^{3}+x_{0}^{2}x_{2}$; or a smooth conic and a secant line, with $n=2$, $d=3$, and $f=x_{1}(x_{1}^{2}+x_{0}x_{2})$. Huh's conjecture. A projective hypersurface with only isolated singularities has polar degree $2$ if and only if it is one of these possibilities after a linear change of homogeneous coordinates. The paper proves this classification, so the conjecture is solved.