Zak's conjecture on cohomological normality of projective varieties

Let $X$ be a non-degenerate projective $m$-dimensional subvariety of $\mathbb{P}^n$, and let $i$ and $j$ be integers with $i\geq 1$ and $j\geq 0$.

Zak's conjecture. 1.

$$H^i(\mathbb{P}^n,\mathcal{I}_X(j))=0$$

whenever

$$i+j<\frac{m}{n-m-1}.$$

2. When

$$i+j=\frac{m}{n-m-1},$$

it should be possible to classify all varieties for which

$$H^i(\mathbb{P}^n,\mathcal{I}_X(j))\neq 0.$$

This conjecture generalizes the question of classifying non-degenerate projective varieties with exceptional failure of $j$-normality. The source presents it as a conjecture of F. L. Zak; no resolution is given here.