Independent products in real L1: asymptotic midpoint convexity without AUC renormings

For a countably branching tree, the closed real L${}^1$ spans of products of independent exponential or Gaussian-square multipliers along its paths have positive averaged asymptotic midpoint moduli and admit no equivalent asymptotically uniformly convex norm. The renorming obstruction holds for every positive nonconstant mean-one multiplier with finite second moment.