We carried out ^ { 12 } CO ( 1 - 0 ) and ^ { 12 } CO ( 2 - 1 ) observations of 21 different regions in the vicinity of M86 , NGC4438 , and along the 120 kpc-long , { H \alpha } -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 ^ { 7 } K intracluster medium ( ICM ) . We targeted { H \alpha } -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 Millimétrique , 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 ^ { -1 } , and with a corresponding { H _ { 2 } } mass of 2 \times 10 ^ { 7 } M _ { \odot } , was detected \sim 10 kpc southeast of the nucleus of M86 , near the peak of its H I emission . We argue that it is possible for this molecular gas either to have formed in situ from H I , or to have been stripped from NGC4438 directly in molecular form . In situ formation is nonetheless negligible for the 7 \times 10 ^ { 6 } M _ { \odot } of gas detected at 12:26:15.9 + 12:58:49 , at \sim 10 kpc northeast of M86 , where no ( strong ) H I emission is present . This detection provides evidence for the survival of molecular gas in filaments for timescales of \sim 100 Myr . An amount equivalent to 5 \times 10 ^ { 7 } M _ { \odot } of { H _ { 2 } } gas that could be lost to the ICM or to neighboring galaxies was also discovered in the tidal tail northwest of NGC4438 . A scenario in which gas was alternatively brought to M86 from NGC4388 was also examined but it was considered unlikely because of the non-detection of CO below or at the H I stream velocities , 2000-2700 km s ^ { -1 } .