Nefness conjecture for the diagonal class on a very general self-product of a curve

Let $C$ be a smooth projective curve of genus $g$ over $\mathbb C$. Denote by $f_1$ and $f_2$ the classes of the fibers of the two projections, and by $\delta$ the class of the diagonal $\Delta$ in $C\times C$. Nefness conjecture. If $g$ is sufficiently large and $C$ has very general moduli, then

$$(1+\sqrt{g})(f_1+f_2)-\delta\in\operatorname{Nef}(C\times C).$$

This is an open problem about the nef cone of self-products of curves; it asks for a boundary nef class of self-intersection zero in a setting where explicit nef classes are difficult to determine.