Minimal splitting-type conjecture for Weyl translates of a line

Let $X$ be the blow-up of $\mathbf{P}^2$ at general points, let $W(X)$ be the Weyl group, and let $C=wL$ for some $w\in W(X)$. Set $d=C\cdot L$, let $m$ be the maximum of $C\cdot E_1,\ldots,C\cdot E_n$, and write $C_i=wE_i$. Let $(a_C,b_C)$ be the splitting type of $p^*\Omega_{\mathbf{P}^2}(1)$ restricted to the normalization of $C$.

Splitting-type minimization conjecture. The pair $(a_C,b_C)$ is the solution $(a,b)$ satisfying

$$a\le b,\qquad \min(m,d-m)\le a\le d-m,\qquad d=a+b,$$

that minimizes

$$\frac{(a-1)(a-2)}{2}+\frac{(b-1)(b-2)}{2}$$

subject to the condition $(**)$ defined in the paper.

The conjecture is motivated by computations of many examples, in which the observed splitting type always minimized the stated quantity subject to $(**)$. It would provide an algorithmic characterization of these splitting types, but is not proved in general.