The Amex pop-up is a mystery. This is pure speculation, but I imagine it could be some sort of customer profitability score being compared to how much the SUB will cost them. That's why you tend to get pop-up on support links, elevated SUBs, and certain cards. Occasionally, there are links where people who are pop-up prone can get past. Maybe then the threshold is set low.
If you haven't gotten the pop-up at all, you might want to take advantage of this time. While you're spending on for the SUB, they might still see you as profitable. I was able to avoid the pop-up for some time while I was consistently getting Amex cards. It was only after I slowed down that I started getting it.