Ideally, the role-holder that accepted the project would recognize the need for additional resources and work with the appropriate role/roles (most likely the Lead Link) to get the additional resources needed to complete the project.
The role-holder should also be able to recognize the importance of the project and prioritize their work accordingly. If there is disagreement between the project owner and the original role with the tension, then the tension-holder should start by making a pitch to the project owner to shift the priority of the project. If this doesn't work, then the tension-holder should ask the Lead Link to evaluate the project and to determine if he/she should ask the project-holder re-prioritize their work to support the project.
Both of these issues could be handle by going role-to-role, or could be added to the agenda of a Tactical meeting.
I hope this helps.