We construct a smooth signed radial weight for which a Grushin Hardy--Rellich inequality holds for every compactly supported smooth test function on the punctured unit gauge ball, but the associated Bessel equation has no positive solution. The principal weight is the constant one and the homogeneous dimension is five. The second weight is specified by the lowest eigenvalue of -d^2/dr^2+r^(-4) on (0,1). A quantitative spectral estimate on a codimension-one subspace yields the explicit positive margin 51/8900, and the Grushin spherical decomposition extends the inequality from radial tests to all tests. This is a counterexample to the remainder-free inequality with an unrestricted signed second weight, not to the forward Bessel-pair theorem or to nonnegative-weight variants. The preprint is unrefereed and makes no absolute priority claim.
