Weyl-line intersection conjecture for effective divisors on $X^4_s$

Let $X^4_s$ be projective four-space blown up at $s$ general points, let $D$ be an effective divisor on $X^4_s$, and let $C$ be a Weyl line, meaning a curve in the Weyl-group orbit of a line through two of the blown-up points. Write $H^1(X^4_s,\mathcal{O}_{X^4_s}(D))$ for the first cohomology group of the associated line bundle.

Weyl-line intersection conjecture. If

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

then

$$D\mathbin{\cdot}C\geq -1$$

for every Weyl line $C$.

The preceding proposition gives a cohomological lower-bound mechanism when a Weyl line has sufficiently negative intersection with $D$, motivating this converse-type necessary condition. The source does not report a resolution.