Slope conjecture for the moduli space of curves

For $g\geq 3$, define the slope of the moduli space of stable curves by

$$s_g:=\inf\{a\in\mathbb{R}_{>0}:|a\lambda-\delta|\neq\emptyset\},$$

where $\delta=\delta_0+\delta_1+\cdots+\delta_{\lfloor g/2\rfloor}$ and $\lambda\in\operatorname{Pic}(\overline{\mathcal{M}}_g)\otimes\mathbb{R}$. Slope conjecture. One has

$$s_g\geq 6+\frac{12}{g+1}$$

for each $g\geq 3$, with equality when $g+1$ is composite. Brill–Noether divisors give the upper bound $s_g\leq 6+12/(g+1)$ when $g+1$ is composite, so the conjecture predicts the optimal value in those cases and a universal lower bound in all genera.