I have this empirical sample, which I drew on the scatter plot. It's got to be unimodal according to theory, and it surely does look like one. A simple way to model it is with bivariate normal. However, the marginals appear to be fat-tailed and slightly skewed, e.g. skewed t-distribution has a similar shape.
Can you suggest candidate distributions for this kind of data?
I fit the Gaussian kernel density estimator for a contour plot just for fun, I don't think this particular kernel would be a good fit because it can't produce fat tails.