Distribution of weighted sum of non-central chi-squared r.v.'s: log-concave?

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)?

• (To be clear: If all the weights are equal, then $X$ is log-concave in both the central and non-central cases, as long as $n\geq 2$.) – Clement C. Feb 3 '19 at 23:50