A Uniform-Distribution Counterexample to Proposed Log-Concave Concentration Extremal Bounds

We give an elementary counterexample to both equalities proposed in Conjecture 1.3 of Pinelisconcerning one-sided concentration of mean-zero, variance-one, log-concave distributions. The centered,standardized uniform distribution has smaller density at the mean than the proposed translated exponentialextremizer and smaller one-sided interval mass. We identify the exact interval-length threshold for thiscomparison and show that the obstruction persists for smooth, strictly positive log-concave densities. Thecounterexample does not challenge the proved universal lower bounds.