Conte simply doesn't want him to leave as he sees Kvaratskhelia as an important part of his system. This is why he is making statements like these.
However Kvaratskhelia made a statement like this after Conte's words.
https://x.com/FabrizioRomano/status/1806098263021416745?t=08Ovz3J5buTjMzeVPQ-u_A&s=19Things are getting heated though.

He doesn't want Conte to talk on his future clearly. Therefore I believe things are still likely to change after Euro 2024. Teams would continue to be interested in him until then.
I mean it wouldn't be expected of him to move during the euros anyway, so of course it will be waited until only after. However, I also think that he is not really going to stay neither, it's obvious that he is aware that it's his time to move to a bigger club and get paid a lot more, there is a cap on how much Napoli could possibly pay him, and if he goes to some other team, lets say a team like Chelsea for example, he would be paid much much more.
So, it is only normal for him to make that decision and not let his manager make that decision. He will request a trade as soon as his euro days are done, and I am sure there are many teams around right now who would want to get a player like him and willing to pay top dollar for him as well.