A desktop system on your network has been compromised. Despite loading different operating systems using different media on the same desktop, attackers appear to have access to that system every time it is powered up and placed on the network. This could be an example of what type of rootkit

Firmware rootkit

Explanation:

From the question, since despite loading different operating systems using different media on the same desktop, and the fact that attackers appear to have access to that system every time it is powered up and placed on the network. This could be an example of firmware rootkit

Firmware rootkits are a threat to ones online safety. In this scenario, the firmware of the computer are targeted by the rootkits and not the operating system and malware are then installed which can destroy the router, or hard drive. This can be used by hackers.