The Nexus Guide says a few relevant things.
The outcome is a set of Sprint Goals that align with the overarching
Nexus Sprint Goal, each Scrum Team's Sprint Backlog and a single Nexus
Sprint Backlog. The Nexus Sprint Backlog makes the work of all Scrum
Team's selected Product Backlog items and any dependencies
All Product Backlog items selected for the Sprint and their
dependencies should be made transparent on the Nexus Sprint Backlog.
A Nexus Sprint Backlog is the composite of Product Backlog items from
the Sprint Backlogs of the individual Scrum Teams. It is used to
highlight dependencies and the flow of work during the Sprint. It is
updated at least daily, often as part of the Nexus Daily Scrum.
The individual Scrum Teams maintain their Sprint Backlogs, which consist of the Product Backlog Items and the work that the Scrum Team has decomposed the Product Backlog Items into. Consider that the Scrum Guide states that the Sprint Backlog "makes visible all the work that the Development Team identifies as necessary to meet the Sprint Goal". That means that a Sprint Backlog contains not only Product Backlog Items, but work that is at a finer granularity than Product Backlog Items.
The Nexus Sprint Backlog contains the Product Backlog Items being worked on by one of the teams plus the subset of work that identifies dependencies between teams. Like the Sprint Backlogs for the teams make the work of the team visible, the Nexus Sprint Backlog makes the cross-team work visible. The cross-team work is the Product Backlog Items that are intended to be delivered and fully integrated at the end of the Sprint, but also any work that requires awareness across the individuals teams to ensure that the work is delivered and fully integrated.
I would say that the primary owner is the Nexus Integration Team. However, if any one on any of the Scrum teams identifies new work that would extend across two or more teams in the Nexus, they would be able to add it to the Nexus Sprint Backlog through the team's representative on the Nexus Integration Team. Consider that the members of the Nexus Integration Team "are often members of the individual Scrum Teams in the Nexus" and that their membership in the Nexus Integration Team "takes precedence over individual Scrum Team membership". That would mean that the members of the Nexus Integration Team are responsible for and accountable for ensuring that the cross-team work items added to the Nexus Sprint Backlog are complete.