I inherited stain just like the one you mention here one time when I took over a church account. The previous cleaner was a part-timer who was a fireman full-time. I'm not saying he didn't know what he was doing, he may have been Jim Martin approved and a bag of chips, but they told me they had been having a problem with this stain for some time. It was a Baptist church, which mean it had a 99.9% chance of being a coffee stain. It was right in the middle of the center aisle of their sanctuary, so they wanted it gone.
On the first cleaning, I treated it with 40 vol. and let it dwell for a bit, then extracted it. Since it was GDC, I COULDN'T USE THE WATER CLAW. So I extracted it really good with my glided wand and put a fan on it to dry while I cleaned the rest of the sanctuary. By the time I got back to it, the stain was starting to show back up. So I repeated the process and finished the job. I came back a couple of days later to check it and it was back, but I could barely see it. I then encapped it and left.
I didn't see it again until 6 or 8 months later when I came back to clean the church again. I noticed it was back again, but only very slightly. I cleaned it like normal along with the rest of the church and it's been gone ever since.
The moral of the story is that sometimes stains like this just take time and repeated cleanings.
But, encap cleaning will almost always give you time to work with it by hiding the stain until you can completely remove it.