Sols and Sols' conjectural bound for sections of rank-two bundles

Let $C$ be a smooth irreducible curve of genus $g$, and let $E$ be a rank-two vector bundle of degree $d$ on $C$. Let $-e$ be the minimum degree of a twist $E\otimes L^{-1}$ having sections, where $L$ ranges over line bundles on $C$; assume that $\mathbb{P}(E)$ is not $C\times\mathbb{P}^1$.

Sols and Sols' conjecture. If

$$-e\leq d\leq 4g-4+e,$$

then

$$h^0(E)\leq \frac{d+e}{2}+1.$$

The conjecture concerns sharp bounds for global sections of rank-two vector bundles over curves and is tied to invariants of the associated ruled surface. The source provides no evidence of a resolution.