That's a difficult question in the sense that there is no level of coordination between departments for them to ultimately know if a supplier is working simultaneously among multiple departments. I would suggest that is in fact happening.
Perhaps, if you'll allow, I'll give a plug for a global vendor performance management framework across the federal government so that information is shared about the performance of suppliers across the entirety of the federal government. It's something I've actively been pushing for.
I know that Public Services and Procurement has been implementing a vendor performance management framework within the department, and it is looking to pilot more ambitious pilots in the future. I think that will allow for information to be shared with other departments and agencies with respect to one supplier that acts with one department and perhaps behaves well or poorly. That would address the concern you've raised.