Each PO must understand what it is they are supposed to do, here we know they are PO in name only.
He who shouts loudest wont work in scrum, its about value and maximizing it.
Oliver has also hit on it, that the SM will be a major part in this, it will be essential.
If they all understood what it is combined they are supposed to be doing would be a start.
This will take some time as they are not even in scrum mode, more likely PM mode.
Could you not apply a visualization room for all the work from all the PO's, where its visible.
When it comes to business value all the PO's would have to come up with valid negotiation points.
Its important wont cut it, its why is it more important than feature x or product y based on data.
No different to sizing when it comes to the team, but at product owner level based on overall value.
Its very clear it wont be easy, as there will have to be trade offs and negotiation by everyone.
If no one is prepared to do this then no matter what option you go for none will work.
This could be a fun way to get them into the swing of things by running a few sessions.
Negotiation and trade offs is going to be key to making it flow between all product owners. http://www.agile42.com/en/agile-coa...alue-game/