Unconventional superconductivity (SC) often develops in magnetic metals on the cusp of static antiferromagnetic (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 SC Cr0.8Ru0.2 alloy (Tc=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 2Δ (the superconducting gap) and a suppression of magnetic spectral weight at energies well above 2Δ. 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.