Nonemptiness conjecture for coherent systems with $n+1$ sections

Let $C$ be a Petri curve of genus $g\ge3$, meaning that the Petri map is injective for every line bundle on $C$. For coherent systems of type $(n,d,n+1)$, define

$$\beta(n,d,n+1):=n^2(g-1)+1-(n+1)\bigl(n+1-d+n(g-1)\bigr),$$

and let $U(n,d,n+1)$ denote the locus of systems whose underlying bundle is stable and which are $\alpha$-stable for all relevant $\alpha$.

$n+1$-section nonemptiness conjecture. If

$$\beta(n,d,n+1)\ge0,$$

then

$$U(n,d,n+1)\ne\emptyset.$$

The source presents this as equivalent, in the stated $m=1$ case, to the strong form of Butler’s conjecture. It is known in many cases and narrowly fails for genus $2$, but remains open in general.