Doron Gepner's Statistics on Words in {1,2,3} is (most probably) Asymptotically Logistic

Doron Gepner’s word statistics, that came up in his research in conformal field theory, is studied and it is conjectured that its scaled limiting distribution is the Logistic distribution. We support this by proving rigorously that the scaled limits of the first twelve moments do indeed converge to those of the Logistic distribution. This is surprising, since Gepner’s statistics is a natural analog of the classical statistics called the number of inversion, that is known to be asymptotically normal.