Gimigliano–Harbourne–Hirschowitz conjecture for effective divisors on blown-up planes

Let $X=X_{2,s}$ be the blow-up of the projective plane at $s$ general points, and let $D\text{ in }\operatorname{Pic}(X)$ be an effective divisor of the specified form. Write $\chi(X,\mathcal{O}_X(D))$ for the Euler characteristic of $\mathcal{O}_X(D))$. A $(-1)$ curve is a smooth, irreducible rational divisor $C$ with $C\cdot C=-1$. Gimigliano–Harbourne–Hirschowitz conjecture. One has

$$\chi(X,\mathcal{O}_X(D))=\dim H^0(X,\mathcal{O}_X(D))$$

if and only if $D\cdot C\geq -1$ for every $(-1)$ curve $C$ on $X$. This is also known as the SGHH conjecture and is equivalent to a conjecture of Segre; its general status is not established in the supplied text.