Speciality conjecture for linear systems on the blow-up of $(\mathbb{P}^1)^3$

Let $Y$ be the blow-up of $(\mathbb{P}^1)^3$ at $r$ points in very general position, and let $Q$ be the divisor in the strict transform of $\mathcal{L}_{(1,1,1)}(1^7)$ described as the image of the strict transform of a quadric through $9$ points of $\mathbb{P}^3$. For a divisor $D$ in the strict transform of $\mathcal{L}_{(d_1,d_2,d_3)}(m_1,\dots,m_r)$, set

$$q(D):=\chi(D|_Q)=(d_1+1)(d_2+1)(d_3+1)-d_1d_2d_3-\sum_{i=1}^7\dfrac{m_i(m_i+1)}{2}.$$

Speciality conjecture. Let $\mathcal{L}:=\mathcal{L}_{(d_1,d_2,d_3)}(m_1,\dots,m_r)$ be a linear system in standard form and let $D$ be a divisor in its strict transform. If $q(D)\leq 0$, then

$$h^0(D)=h^0(D-Q).$$

If $q(D)>0$, then $D$ is special if and only if $m_1>d_n+1$ and $D$ is fiber non-special.

This conjecture is equivalent, via the small modification $\phi$, to Conjecture 6.3 of Laface and Ugaglia. The statement concerns the relationship between the divisor $Q$, the numerical invariant $q(D)$, and speciality of linear systems on the blow-up; the supplied text gives no evidence that it has been resolved.