We carried out CO(1-0) and CO(2-1) observations of 21 different regions in the vicinity of M86, NGC4438, and along the 120 kpc-long, Ha-emitting filamentary trail that connects them, aiming to test whether molecular gas can survive to be transferred from a spiral to an elliptical galaxy in Virgo's 10^7K intracluster medium (ICM). We targeted Ha-emitting regions that could be associated with the interface between cold molecular clouds and the hot ionized ICM. The data, obtained with the 30m telescope of the Institut de Radioastronomie Millimetrique, led to the detection of molecular gas close to M86. CO gas with a recession velocity that is similar to that of the stars, -265 km/s, and with a corresponding H2 mass of 2*10^7 M_sun, was detected ~10 kpc south east of the nucleus of M86, near the peak of its HI emission. We argue that it is possible for this molecular gas either to have formed in situ from HI, or to have been stripped from NGC4438 directly in molecular form. In situ formation is nonetheless negligible for the 7*10^6 M_sun of gas detected at 12:26:15.9+12:58:49, at ~10 kpc north east of M86, where no (strong) HI emission is present. This detection provides evidence for the survival of molecular gas in filaments for timescales of ~100 Myr. An amount equivalent to 5*10^7 M_sun of H2 gas that could be lost to the ICM or to neighboring galaxies was also discovered in the tidal tail north-west of NGC4438. A scenario of gas being alternatively brought to M86 from NGC4388 on its south was also examined but it was considered unlikely due to the non detection of CO below or at the HI stream velocities, 2000-2700 km/s.