Extended Bertram–Feinberg–Mukai conjecture for canonical coherent systems

Let $C$ be a general curve of genus $g$, let $K$ be its canonical bundle, and let $G(\alpha;2,K,k)$ denote the moduli space of $\alpha$-stable rank-two coherent systems with determinant $K$ and $k$ sections. Let $U(2,K,k)$ denote the corresponding locus with stable underlying bundle, and set

$$\beta(2,K,k):=3g-3-\frac{k(k+1)}2.$$

Extended Bertram–Feinberg–Mukai conjecture. One has

$$\beta(2,K,k)<0\Rightarrow G(\alpha;2,K,k)=\emptyset\quad\text{for all }\alpha>0,$$

and

$$\beta(2,K,k)\ge0\Rightarrow U(2,K,k)\ne\emptyset\quad\text{for all }\alpha>0.$$

This extends the canonical rank-two Brill–Noether conjecture from bundles to coherent systems and all positive stability parameters. The source states it as an open problem.