Converts a single cis double bond at position 12 of linoleate incorporated into phosphatidylcholine into conjugated 11-trans and 13-cis double bonds (PubMed:12464604). 16:2(9Z,12Z), 18:3(9Z,12Z,15Z) and 18:2(9Z,12Z) are substrates for the conjugase to form trans-Delta(11) and cis-Delta(13) double bonds (PubMed:12464604). No activity on the cis-Delta(9) double bonds of oleic and palmitoleic acids (PubMed:12464604). a (9Z,12Z)-octadecadienoyl-containing glycerolipid + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = a (9Z,11E,13Z)-octadeca-9,11,13-trienoyl-containing glycerolipid + 2 Fe(III)-[cytochrome b5] + 2 H2O Lipid metabolism; polyunsaturated fatty acid biosynthesis. The histidine box domains may contain the active site and/or be involved in metal ion binding. Belongs to the fatty acid desaturase type 1 family.