A strict four-row permanent inequality and permutation moments

We prove a permanent inequality with exponent strictly below two for laws on S${}_4$ sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on $N=2^d$ cards to total-variation distance tending to zero from uniform as $N\to \infty$. Thus the mixing time has the optimal order $O(\log N)$ in physical shuffles.