Converts a single cis double bond at position 12 of linoleate incorporated into phosphatidylcholine into conjugated 11-trans and 13-cis double bonds (PubMed:12354116, PubMed:12464604, PubMed:25000918). Produces punicic acid (18:3(9Z,11E,13Z)) from linoleic acid and conjugated octadecatetraenoic fatty acid from gamma-linolenic acid (PubMed:12354116, PubMed:25000918). No activity with cis- and trans-vaccenic acid, alpha-linolenic acid or homo-gamma-linolenic acid (PubMed:12354116). 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. Truncated C-terminus.