Huh's finiteness conjecture for fixed polar degree

Let $k>2$, let $V\subset\mathbb{P}^{n}$ be a projective hypersurface with only isolated singular points, and let $d=\deg V$. Huh's finiteness conjecture. For sufficiently large $n$ and $d$, there is no such hypersurface $V$ of polar degree $k$. This is presented as a general finiteness principle; the supplied text does not state a resolution for arbitrary $k>2$.