In HP and the Deathly Hallows, when hermione is talking about means of destroying a Horcrux, she says the below sentence about destroying a Horcrux

It has to be something so destructive that the horcrux can't repair itself

And then she also states

Our problem is that there are very few substances as destructive as Basilisk venom, and they're all dangerous to carry around with you. That's a problem that we're going to have to sove, though, because ripping, smashing or crushing a Horcrux won't do the trick

Now as you have quoted Lupin

I suppose there must be some kind of mouth under there, because they clamp their jaws upon the mouth of the victim and – and suck out his soul.

If we can assume that sucking the soul is somewhat similar to it being ripped out from the body, then we can say that the dementor kiss will not destroy the horcrux

In HP and the Deathly Hallows, when Hermione is talking about means of destroying a Horcrux, she says the following:

"It has to be something so destructive that the Horcrux can't repair itself."

And then she also states

"Our problem is that there are very few substances as destructive as Basilisk venom, and they're all dangerous to carry around with you. That's a problem that we're going to have to solve, though, because ripping, smashing or crushing a Horcrux won't do the trick."

Now, as you have quoted Lupin

"I suppose there must be some kind of mouth under there, because they clamp their jaws upon the mouth of the victim and – and suck out his soul."

If we can assume that sucking the soul is somewhat similar to it being ripped out from the body, then we can say that the Dementors' Kiss will not destroy a Horcrux.