Clifford-type bound for stable coherent systems on quintic surfaces

Let $S\subset\mathbb{P}^3$ be a smooth quintic surface and let $h=c_1(\mathcal{O}_S(1))$. A coherent system on $S$ is a morphism $(\mathcal{O}_S^{\oplus R}\to F)$; let $\mu'$ denote the slope stability function used in the source.

Clifford-type bound conjecture. For every $\mu'$-stable coherent system $(\mathcal{O}_S^{\oplus R}\to F)$ with

$$R=2\operatorname{rank}(F)>0,$$

one has

$$\frac{c_1(F)\cdot h}{R}>\frac{5}{8}+\frac{c}{b}=1.3818\cdots.$$

This conjecture is intended to provide the inequality needed for the double-tilting construction of a Gepner-type stability condition on the corresponding matrix-factorization category. The source does not prove the bound.