I wouldn't be surprised if they are already chucking large sums of money at a search effort while the AIBs argue about whose jurisdiction it is.
Why would there be arguing? The rules are quite clear:
Denmark has the lead (state of occurrence)
Rights of participation:
USA (design and manufacture engine)
France (design and manufacture airframe, operator and registry.
Everyone else provides assistance as requested. It seems obvious that Canada plays a major part as that's where the aircraft is.
Given the scale of the investigation and that, other than being under the flight path they have nothing to do with it, Denmark may choose to hand it over to one of the others. But that's their choice to make.