As someone loathe to use chemicals, I thought perhaps that the tenacity of the plant was overhyped, so I simply pulled the stalk out of the ground. Over the next several months, I'd notice 3-4 new stalks popping up each week over a roughly 15 foot radius area. That continued through the next spring when I finally decided enough was enough and got the glysophate.
From June through winter (I'm in the northeast US), I'd let a plant grow for a week or two and then spray it with the glysophate and leave the plant in place. I did this for the entirety of the following year as well. Now, in year 3, I finally have no more shoots popping up, and can't begin to imagine what managing a true infestation looks like.