A Centering Counterexample for Prekopa-Leindler Stability

A conjecture printed in the 2009 Oberwolfach report on convex geometry proposes quantitative Prekopa-Leindler stability in the unshifted L1 distance of probability densities with zero mean. We give an explicit counterexample in every dimension. Strictly positive real-analytic densities with mean zero and covariance I_n have L1 distance at least 11/8, while an admissible midpoint majorant has integral 1+2k/(1+k^2), for every integer k>=3. The same densities admit majorants with vanishing deficit for every fixed interpolation parameter in (0,1), uniformly on compact subintervals. Small remote Gaussian components cancel the barycenters without aligning the main mass. An explicit translation makes the distance at most 2/(1+k^2), consistent with established stability modulo translation. This English preprint addresses the literal UnsolvedMath OWR-4136-004 formulation; it does not contradict log-concave or translated stability theorems. The elementary argument is AI-assisted and self-audited, not independently peer reviewed or formally verified. No historical-priority claim is made.