Diaconis–Perlman unique crossing conjecture for weighted gamma sums

Let $X_1,\ldots,X_n$ be independent and identically distributed gamma$(\alpha,1)$ random variables, and let $F_\theta$ denote the distribution function of $\sum_{i=1}^n \theta_iX_i$ for a nonnegative weight vector $\theta=(\theta_1,\ldots,\theta_n)$. For weight vectors, write $\eta\prec\theta$ when $\eta$ is majorized by $\theta$.

Diaconis–Perlman unique crossing conjecture. If $\eta\prec\theta$, but $\theta$ is not a permutation of $\eta$, then $F_\eta(x)-F_\theta(x)$ changes signs exactly once, from $-$ to $+$, as $x$ increases from $0$ to $\infty$.

The conjecture concerns the ordering of distribution functions for convolutions of identically distributed gamma random variables under majorization of the weights. It is disproved when the shape parameter satisfies $\alpha<1$, while it is proved when $\alpha\geq 1$; thus the conjecture is not valid in its original unrestricted form.