SHGH-type conjecture for negative curves on the Klein blowup

Let $X_\mathcal K$ be the blowup associated with the Klein line configuration, let $H$ be the pullback of a line, and let $E_4$ and $E_3$ denote the exceptional divisors over the quadruple and triple points. For

$$D=dH-m_4E_4-m_3E_3,$$

write $T_d(-m_4E_4-m_3E_3)$ for the corresponding linear system. SHGH-type conjecture. If $D\mathbin{.}C\geq 0$ for every $G$-invariant, $G$-irreducible curve $C$ of negative self-intersection with degree less than $d$, then

$$\operatorname{edim}\bigl(T_d(-m_4E_4-m_3E_3)\bigr)=\dim\bigl(T_d(-m_4E_4-m_3E_3)\bigr).$$

Computer calculations suggest that equality holds on the Klein blowup unless a negative curve lies in the base locus, but the conjecture is otherwise open.