Direct imaging (DI) surveys suggest that gas giants beyond 20 AU are rare around FGK stars. However, it is not clear what this means for the formation frequency of Gravitational Instability (GI) protoplanets due to uncertainties in gap opening and migration efficiency. Here we combine state-of-the-art calculations of homogeneous planet contraction with a population synthesis code. We find DI constraints to be satisfied if protoplanet formation by GI occurs in tens of percent of systems if protoplanets `super migrate to small separations. In contrast, GI may occur in only a few percent of systems if protoplanets remain stranded at wide orbits because their migration is `quenched by efficient gap opening. We then use the frequency of massive giants in radial velocity surveys inside 5 AU to break this degeneracy - observations recently showed that this population does not correlate with the host star metallicity and is therefore suspected to have formed via GI followed by inward migration. We find that only the super-migration scenario can sufficiently explain this population whilst simultaneously satisfying the DI constraints and producing the right mass spectrum of planets inside 5 AU. If massive gas-giants inside 5 AU formed via GI, then our models imply that migration must be efficient and that the formation of GI protoplanets occurs in at least a tens of percent of systems.
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