It's known that for a random vector $(X_1,\dots,X_n)\in \mathbb{R}^n$ with a log-concave distribution, any subvector has a long-concave distribution. I'm wondering if there are any results about its converse, in particular, when $(X_1,\dots,X_n)$ is isotropic, $X_i$'s are identically distributed (but not necessarily independent) and each $X_i$ is log-concave.

Is this true when $n=2$? That is,

Suppose that $X$ and $Y$ are identically distributed subject to some log-concave distribution with $E[X]=E[Y]=0$, $E[X^2]=E[Y^2]=1$ and $E[XY]=0$. Is the joint distribution of $(X,Y)$ log-concave?