We propose a stringent observational test on the formation of warm Jupiters (gas-giant planets with 10 d <~ P <~ 100 d) by high-eccentricity (high-e) migration mechanisms. Unlike hot Jupiters, the majority of observed warm Jupiters have pericenter distances too large to allow efficient tidal dissipation to induce migration. To access the close pericenter required for migration during a Kozai-Lidov cycle, they must be accompanied by a strong enough perturber to overcome the precession caused by General Relativity (GR), placing a strong upper limit on the perturbers separation. For a warm Jupiter at a ~ 0.2 AU, a Jupiter-mass (solar-mass) perturber is required to be <~ 3 AU (<~ 30 AU) and can be identified observationally. Among warm Jupiters detected by Radial Velocities (RV), >~ 50% (5 out of 9) with large eccentricities (e >~ 0.4) have known Jovian companions satisfying this necessary condition for high-e migration. In contrast, <~ 20 % (3 out of 17) of the low-e (e <~ 0.2) warm Jupiters have detected additional Jovian companions, suggesting that high-e migration with planetary perturbers may not be the dominant formation channel. Complete, long-term RV follow-ups of the warm-Jupiter population will allow a firm upper limit to be put on the fraction of these planets formed by high-e migration. Transiting warm Jupiters showing spin-orbit misalignments will be interesting to apply our test. If the misalignments are solely due to high-e migration as commonly suggested, we expect that the majority of warm Jupiters with low-e (e <~0.2) are not misaligned, in contrast with low-e hot Jupiters.
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