The O-O conjecture for splitting types of primitive covers

Let $\alpha\colon C\to\mathbb{P}^1$ be a degree $k$ cover, and let $\vec{a}=(a_1,\ldots,a_{k-1})$ be defined by

$$(\alpha_*\mathcal{O}_C)^{\vee}=\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(a_1)\oplus\cdots\oplus\mathcal{O}_{\mathbb{P}^1}(a_{k-1}).$$

The tuple $\vec{a}$ is the splitting type, or the scrollar invariants, of the cover. O-O conjecture. There exists a primitive cover $C\to\mathbb{P}^1$ of splitting type $\vec{a}$ if and only if $a_{i+j}\leq a_i+a_j$ for all $i,j$. The conjecture holds in low degree by parametrizations of covers, and a positive proportion of the admissible splitting types is known in larger degree, but the full assertion remains open.