Can the shellshock bug be expoloited to run a command as a privileged user?

I am curious to know if the following command could be executed on a system/server with the Shellshock bug: curl -H "User-Agent: () { :; }; sudo /bin/eject" http://example.com/ (This code is an elaborated version of an example here: http://blog.cloudflare.com/inside-shellshock/) From what I understand the bug would allow one to inject code into the system but not necessarily execute code with elevated privileges. If this is not possible is there another way that code could be injected to give a hacker root privileges? No examples necessary, I am simply curious as to the extent of damage this bug could potentially cause. **I have no malicious intent, I ask out of curiosity.