Based on what I recall about Atlanta moving to Winnipeg, the only reason it happened was because the NHL and Atlanta ownership had thoroughly exhausted any and all possibilities of keeping the team in Atlanta. Once it was accepted that there were no potential owners who were able to keep the team in Atlanta, that's when the NHL allowed ownership bids that included relocation to jump in.
Plus, if Eugene wants to relocate, what's he going to do? Relocate to a city that already has a rink? So basically Quebec? But, he won't be the owner of the rink, so that's a pain for him to pay rent, not own the rink, and then no be able to split the income so he can cry poor.
That makes the other option being that he'd have to negotiate with the new city and state/province to build a new arena in the new city. If he's lucky they'd cover what, 50% of the cost? So he'd need to come up with roughly $250M in cash for the new rink and I would guess $75M for the relocation fee. Add that $325M to his already burdensome debt of around $150M, and you've got a guy who's now broke as a joke. Plus, cities and provinces/states never just give up the money. It's always tied to some sort of repayment plan, either by special tax, or the team pays back a certain amount over a 20 year span. So that could also potentially eat into his profits.
Good luck getting approval from the NHL for all of that.
I guess he could sell CTC and surrounding land, but I don't think that would cover his costs.
It's not happening while he's the owner of this team.