As far as I know, there are robust regression methods for outliers in response $Y$ and heavy-tailed error $\epsilon$. The settings for the design matrix (predictor) $X$ is either fixed design or sub-exponential like Normal distribution. What if the design matrix $X$ is random design and the distribution is heavy-tailed? Is there any advanced methods apart from truncation on $X$ or $\log X$?
One method/idea is resistant regression. The idea is to order the squared residuals, rejecting some percent of the largest ones, and then minimize the sum of squared kept residuals. That is highly nonlinear and nonconvex, and one idea for the optimization is genetic algorithms.
One implementation is
lqs in R package
MASS, see the companion book by Venables & Ripley. For code example see How to get summary statistics from "resistant regression" - lqs - in R?