Nothing concrete, but its widely speculated that he'll join the team after this season.
The lack of a contract extension at this point would tend to point towards him not extending with SKA, and the goalie coach of SKA recently said he expects him to leave after this season. He's also not one of these players, like Kaprizov or Sorokin, who has made comments against his NHL team. He's really not said anything that would cause controversy, relating to the NHL. He's said the usual he wants to play in the NHL eventually.
He's also said his idol is Lundqvist, so that definitely helps get him over to the team. And then there are enormous Russian ties within the team that are around his age, whether its Buchnevich, Georgiyev or other prospects like Kravtsov and Rykov. He plays with Rykov at SKA, and he's played with Kravtsov for the Russian National Team. After Kravtsov was drafted, Shestyorkin congratulated him on instagram, and they talked about playing on the same team as each other eventually. He's played with Buchnevich a lot, and based on their instagram accounts, they were friends years ago. Not sure if they still are. Maybe so, although they don't comment on each others photos as much. He might also know Georgiyev, although I'm not sure.
But certainly a lot of connections within the team that will benefit getting him over to the team. The only thing I'd be unsure about here is that he's very good friends with Sorokin, and the Isles obviously are having issues getting him to join. Is Sorokin going to convince him to not move to NA next season?
As of now, fans are operating under the assumption that he'll join after one more year. No one knows for sure, so we'll have to wait until he makes his decision.