The ample-cone conjecture for symmetric products of very general curves

Let $C$ be a smooth curve of genus $g\ge 2$. Its second symmetric product $C^{(2)}$ has numerical divisor classes $x$ and $\delta$, where $x$ is the class of $\{p+q:q\in C\}$ and $\delta$ is the class of the diagonal. Define

$$\tau(C)=\inf\{s>0:(s+1)x-(\delta/2)\text{ is ample}\}.$$

Ample-cone conjecture. If $C$ is a very general curve of genus $g\ge 4$, then

$$\tau(C)=\sqrt{g}.$$

This asserts that the non-diagonal boundary of the ample cone has zero self-intersection, making the nef and ample cones of $C^{(2)}$ as large as possible. It is known when $g$ is a perfect square, but remains open in general.