Unconventional superconductivity (SC) often develops in magnetic metals on the cusp of static anti-ferromagnetic (AFM) order where spin fluctuations are strong. This association is so compelling that many SC materials are labeled as unconventional by proximity to an ordered AFM state. The Cr-Ru alloy system possesses such a phase diagram [see Fig. 1(a)]. Here we use inelastic neutron scattering to show that spin fluctuations are present in a SCCr0.8Ru0.2 alloy (T-c = 1.35 K). However, the neutron spin resonance, a possible signature of unconventional SC, is not observed. Instead, data indicate a spin gap of order 2 Delta (the superconducting gap) and a suppression of magnetic spectral weight at energies well above 2 Delta. The suppression decreases the magnetic exchange energy, suggesting that low energy spin fluctuations oppose the formation of SC. In conjunction with other experimental evidence, a possible scenario is that conventional SC sits on the cusp of AFM order in Cr-Ru alloys.