They would probably be able to do it if the listed prices were fixed, but they provided a personalized discount that would vary depending on who you are.
This is already how the app works and indeed pretty much every form of price discrimination runs this way. It's just that that the "who you are" part often isn't as obvious because you'll just encounter a coupon code and not realize that by virtue of you seeing it you're part of a specific demographic they're targeting.
But the reverse of raising prices beyond menu price depending on who you are will get pitchforks.
But the reverse of raising prices beyond menu price depending on who you are will get pitchforks.