Let $n\geq 2$, and $X_1,\dots,X_n$ be independent non-central r.v.'s, where $X_i \sim \chi^2(\delta_i)$; and $w_1,\dots, w_n > 0$.

Letting $$X \stackrel{\rm def}{=} \sum_{i=1}^n w_i X_i$$ is it true that the distribution of $X$ is log-concave? (I.e., that its pdf is a log-concave function.)

To begin with, is it even true for $X_1,\dots,X_n$ being *central* chi-square r.v.'s? If not, under which assumptions on the weights does it hold (since certainly it is true if all weights are equal)?