A Counterexample to a Homogeneous Fractional Hardy Conjecture on the Heisenberg Group

A conjecture in the 2023 Oberwolfach report on hypoelliptic operators proposes replacing the homogeneous weight in a fractional Hardy inequality by the power of the standard Heisenberg homogeneous norm while retaining the displayed constant. We give a complete counterexample to this replacement. On the first Heisenberg group, the explicit real function rho^(-1) sin^2(log(rho)/3), supported where 1 < rho < exp(3 pi), violates the proposed inequality for every fractional order 99/100 <= s < 1. Its fractional Rayleigh quotient is less than 273/284, whereas the proposed constant is greater than 98/101. The proof combines an exact radial energy calculation, elementary Gamma-ratio bounds and spectral concavity, keeping the conformal fractional operator distinct from the ordinary spectral power. A single smooth compactly supported witness also exists for the whole interval. This four-page English preprint addresses UnsolvedMath OWR-13750335-005. It does not rule out a smaller Hardy constant or assert failure for all smaller orders. Classical horizontal-gradient obstructions are credited. The work is AI-assisted and self-audited, not independently peer reviewed or formally verified; bounded literature searches do not certify absolute priority.