Decreasing-density conjecture for quantile mixtures

Let $n$ be a positive integer, let $\boldsymbol{F}=(F_1,\dots,F_n)\in\mathcal M_D^n$, and let $\Lambda\in\mathcal Q_n$. Write $\Lambda\otimes\boldsymbol{F}$ for the corresponding mixture of the marginal distributions, and let $\mathcal D_n(\boldsymbol{F})$ denote the set of distributions generated by quantile mixtures from $\boldsymbol{F}$. The notation $F^\lambda$ denotes the distribution obtained by scaling the quantile function of $F$ by $\lambda$, $\boldsymbol\gamma\prec\boldsymbol\lambda$ denotes majorization, and $\Delta_n$ is the simplex of weight vectors.

Decreasing-density conjecture. For $\Lambda\in\mathcal Q_n$ and $\boldsymbol{F}\in\mathcal M_D^n$, one has

$$\mathcal D_n(\boldsymbol{F})\subset\mathcal D_n(\Lambda\otimes\boldsymbol{F}).$$

The following are weaker versions:

1. For $F\in\mathcal M_D$ and $\boldsymbol\lambda,\boldsymbol\gamma\in\mathbb R_+^n$, if $\boldsymbol\gamma\prec\boldsymbol\lambda$, then

$$\mathcal D_n(F^{\lambda_1},\dots,F^{\lambda_n})\subset\mathcal D_n(F^{\gamma_1},\dots,F^{\gamma_n}).$$

2. For $F_1,\dots,F_n\in\mathcal M_D$,

$$\mathcal D_n(F_1,\dots,F_n)\subset\mathcal D_n(F,\dots,F),$$

where

$$F^{-1}=\frac{1}{n}\sum_{i=1}^nF_i^{-1}.$$

3. For $F\in\mathcal M_D$ and $(\lambda_1,\dots,\lambda_n)\in\Delta_n$,

$$\mathcal D_n(F^{n\lambda_1},\dots,F^{n\lambda_n})\subset\mathcal D_n(F,\dots,F).$$

These questions arise because quantile-mixture classes are generally not comparable. The paper identifies decreasing densities as a natural setting inspired by the scaling theorem, but does not resolve the conjecture or its weaker forms.