The slope conjecture for effective divisors on the moduli space of stable curves

Let $\overline{\mathcal M}_{g}$ be the Deligne–Mumford moduli space of genus $g$ stable curves. For an effective divisor $D=a\lambda-\sum_{i=0}^{\lfloor g/2\rfloor}b_i\delta_i$ with $a,b_i>0$, define its slope by

$$s(D)=\frac{a}{\min\{b_i\}}.$$

The Brill–Noether bound is $6+\frac{12}{g+1}$. The slope conjecture. The Brill–Noether bound provides a lower bound for slopes of effective divisors on $\overline{\mathcal M}_{g}$, and the bound is sharp if and only if $g+1$ is composite.

The conjecture was proposed because the Brill–Noether divisors have slope equal to the Brill–Noether bound, independently of the choice of $d$ and $r$. It is false: the source explicitly states that the slope conjecture is disproved.