The conjecture on asymptotically special nef divisors on blow-ups of projective space

Let $X^n_r$ be the blow-up of $\mathbb P^n$ at $r$ points in very general position. A nef divisor on $X^n_r$ is asymptotically special if it has the asymptotic property considered in the paper.

Conjecture on asymptotically special nef divisors. The blow-up $X^n_r$ does not contain nef divisors which are asymptotically special.

This is proposed as a possible higher-dimensional generalization of conjectures for blow-ups of $\mathbb P^2$; its status is not established in the supplied text.