The fixed-determinant coherent-systems dimension conjecture

Let $C$ be a smooth projective curve of genus $g$, let $r,d,k$ be integers, let $\mathcal{L}$ be a line bundle on $C$ of degree $d$, and let $G(\alpha;r,\mathcal{L},k)$ denote the moduli space of coherent systems of type $(r,d,k)$ with determinant $\mathcal{L}$. Fixed-determinant dimension conjecture. Every irreducible component $X$ of $G(\alpha;r,\mathcal{L},k)$ has dimension

$$\dim X\ge\beta(r,d,k)-g+\binom{k}{r}h^1(\mathcal{L}).$$

This strengthens the general lower bound by incorporating the contribution from the fixed determinant; it agrees with the rank-one calculation and with the canonical rank-two case discussed immediately beforehand. Its status is not resolved in the supplied text.