Ok I've tracked it down, sadly not a solution but the reason. Derek and others are right. The piping is probably made from viscose which means the dye will be unstable in water or anything above PH 6/7, particularly on Damask suites (that's what rang the bell) It can be stabilised with a low acidic treatment, I used to use fibre bond PH about 3 if I recall, but have not needed it for years, then cleaning with Fib Fab rinse or similar. You can test with a your normal prespray but if it has piping i'd start with ready to use fib fab rinse and certainly never use high PH treatments. Apply solution to a cotton bud and test at back of cusion and keep the area as small as possible because it could stick out like a sore thumb! If there is any doubt don't wet clean and do similar test with OMS, dye can still be unstable so take much care. My old guvnor once clean one by using dry cleaning solvent as barrier before carefully wet cleaning it, two things, 1 he was genius and 2 had kahuna's as big as grapefruits. Not for the faint hearted. BTW was this an old suite? I have not seen one for a while. Sorry it does not offer a solution as it does sound like a dye bleed from viscose in the piping. With the high PH solutions you've tried already acetic acid might take it back. If it looks like a claim you will have nothing to lose.
Cheers and good luck
Simon