Vanishing and global-generation conjecture for divisors on the resolved fourfold

Let $X^4_8$ be projective four-space blown up at eight general points. Let $\widehat{X^4_8}$ be obtained by blowing up the points, Weyl lines, and Weyl planes in the base locus of an effective divisor $D$, and let $\widetilde{D}$ be its proper transform. Let $C$ range over Weyl curves, and let $\operatorname{wdim}(D)$ denote the Weyl expected dimension defined from the Euler characteristic and the multiplicities of Weyl cycles in the base locus.

Vanishing and global-generation conjecture. For every effective divisor $D$ on $\widehat{X^4_8}$:

1. If $D\mathbin{\cdot}C\geq -1$ for every Weyl curve $C$, then

$$H^1(X^4_s,\mathcal{O}_{X^4_s}(D))=0.$$

2. One has

$$h^0(D)=\operatorname{wdim}(D)+\sum_{r=1}^{3}(-1)^{r+1}h^r(\widetilde{D}).$$

3. For every $r\geq1$,

$$h^r(\widetilde{D})=0.$$

4. Moreover, $\widetilde{D}$ is globally generated on $\widehat{X^4_8}$.

This packages the source's stronger conjectural claims about cohomology, the Weyl expected dimension, and global generation. The source does not report a resolution.