The unimodal dispersion comparison for Cramér and average value metric distances

Let $F$ and $G$ be unimodal distributions in the sense that their quantile functions are differentiable almost everywhere, with derivatives $(F^{-1})'$ and $(G^{-1})'$ decreasing for $\alpha < \frac12$ and increasing for $\alpha > \frac12$. Write $\operatorname{CD}$ for the Cramér distance, $\operatorname{AVM}$ for the average value metric, and $\operatorname{Disp}_+^D$ and $\operatorname{Disp}_-^D$ for the positive and negative dispersion components of a distance $D$. Unimodal dispersion comparison conjecture. It may hold that

$$\frac{\operatorname{Disp}_+^{\operatorname{CD}} + \operatorname{Disp}_-^{\operatorname{CD}}}{\operatorname{CD}(F,G)} \; \leq \; \frac{\operatorname{Disp}_+^{\operatorname{AVM}} + \operatorname{Disp}_-^{\operatorname{AVM}}}{\operatorname{AVM}(F,G)}.$$

The inequality would extend the established comparison for normal distributions to possibly asymmetric unimodal distributions. It is motivated by the fact that the comparison typically holds in practice, although counterexamples exist for arbitrary symmetric distributions; its validity for the stated unimodal class remains open.