I'd say you're right and that the majority of the hockey world expects Barzal to be getting 9+ million per on his next contract.
What Lou privately thinks about it, and what he may be planning with respect to Barzal moving forward, is something we'll never find out about - not publicly in the media or unless the very unlikely event of a trade takes place. But Lou will see to it that Barzal gets the money necessary to keep him our (satisfied) asset for as long as possible.
As others have mentioned, there is no way Lou can afford to take the huge step backwards that losing Barzal would be so shortly before Belmont.
At least one would certainly have to think that.
LEE, EBERLE, and NELSON:
This is not directed at you per se, but if there are people out there who believe these three wouldn't have gotten the money they ended up with - perhaps even better - on the free market, then nobody here can help you. I'd only suggest that if you think that way, you don't go waving that train of thought around people who are in the know about the NHL market. As Crew mentioned above, if there are GMs out there who are giving the Jeff Skinners of the world 9+ MM per, then there should be no doubt that our boys could have gotten as much as they did for resigning here - or more - on the free market as well.
In light of having 6 UFAs last summer (four of which were critically important to what the team had achieved), a situation where the team captain could walk for the second summer in a row, and the desire to go for the biggest fish in the UFA pond (who'd have been placed right in front of Lee's nose as the #1 LW), Lou did nothing short of a masterful job in locking down Nelson and Eberle to extremely market-fair deals before being the biggest bidder for Panarin while keeping Lee on stand-by, only to re-up the captain to a deal that he'd surely have gotten elsewhere.
Sure, Lee's deal looks like a deal that may become very ugly towards its conclusion, but Lou couldn't allow this team to take a step backwards on paper after what was established last season. He tried to make it considerably better, but then managed to at least keep it the same. At least with respect to the skaters.
Goaltending is something different, although I'm pretty sure that if Sorokin ultimately signs in time for next season, then the Varlamov signing will be "completed". And I think Lou and Korn were very ready to see goaltending take a slight step back in order to get Sorokin over here, but that all still has to play out...
In any case, Lou will no doubt do what's necessary in keeping Barzal as the cornerstone of this team - as long as some crazy GM doesn't come along offering Mat 12 MM plus per year.
Are there any out there who would/could?
