Slope quotient principle for effective divisors on the moduli space of curves

Let $s_g$ be the slope of $\overline{\mathcal{M}}_g$, and let $s(D)$ denote the slope of an effective divisor $D\equiv a\lambda-\sum_{i=0}^{[g/2]}b_i\delta_i$. Slope quotient principle. For every genus $g$, the following statement should hold: there exists $\epsilon_g>0$ such that any effective divisor $D$ on $\overline{\mathcal{M}}_g$ satisfying

$$s_g\leq s(D)\leq s_g+\epsilon_g$$

has $s(D)=a/b_0$, equivalently $b_0\leq b_i$ for all $i\geq 1$. The theorem establishing this assertion is stated in the paper for $g\leq 23$, while the displayed sentence conjectures its validity in arbitrary genus.