The essentially large divisor conjecture for big divisors in general position

Let $q\geq 1$ and $r\geq q+2$ be integers. Let $X\subseteq \mathbb{P}^\ell$ be a nonsingular projective variety of dimension $q$, defined over $K$. Let $D=\sum_{i=1}^r D_i$ be an effective divisor, defined over $K$, on $X$ such that each $D_i$ is big and the divisors $D_i$ are in general position. Essentially large divisor conjecture. Then $D$ is essentially large.

The preceding theorem establishes the conclusion under the additional codimension condition $2q-\ell\geq 2$. The conjecture asserts that the bound $r\geq q+2$ remains sufficient without any restriction on the codimension of $X$.