An example of a non-log-concave distribution where the difference has a log-concave density

By the Prékopa-Leindler inequality, the difference $X-X^{\mathrm{\prime }}$ has a log-concave density provided that $X$ has a log-concave density and $X,X^{\mathrm{\prime }}$ are independent and identically distributed. We prove that the opposite direction does not always hold true by giving an explicit example.