Mohan Kumar's affineness conjecture for algebraic manifolds

Let $Y$ be an algebraic manifold of dimension $n$, let $X$ be a completion of $Y$, and let $D$ be an effective boundary divisor with support $X-Y$. For integers $i,j\geq 0$, write $H^i(Y,\Omega^j_Y)$ for the cohomology of the sheaf of differential $j$-forms on $Y$, and let $\kappa(D,X)$ denote the $D$-dimension. Mohan Kumar's affineness conjecture. The variety $Y$ is affine if and only if

$$H^i(Y,\Omega^j_Y)=0\quad\text{for all }j\geq 0,\ i>0,\qquad\text{and}\qquad \kappa(D,X)=n.$$

The statement extends the corresponding threefold criterion, which the source says was proved there, to algebraic manifolds in arbitrary dimension. Its status in higher dimension is left open in the source.