Right tail of log-concave densities. Consider any density $g$ with support $(a,b)$ and $-\infty<a<b<+\infty$.
Is it possible to have $g$ log-concave on $(a,b)$ even if $\lim_{q\uparrow b}g(q)=+\infty$?
An (1995, 1998) proves that the right tail of a log-concave density is at most exponential, but the proof seems to assume $(a,b)=\mathbb{R}$. It should hold for, say, $(a,b)=(0,1)$, right?

If$a,b$ are finite then random variables distributed according to $g$ are bounded. Bounded random variables are subgaussian. Subgaussian random variables are subexponential. So then using the claim cited in your question, yes it is possible.
