Here is a challenge problem. I have no personal interest in it, but this was posted at https://mathoverflow.net/questions/303936/concavity-of-log-1-fracx-11x-1x-2-2x-3-4-1-fracx-21x-1-4x-2/303948#303947
This appears to be concave for any n, and is concave for particular values of n per Maple. Is there a conic formulation? It seems tough. Note that if formulated as a sum of log terms, each log term is NOT concave individually, so it seems especially tricky.