Rather strange edge case here, but I am unsure what the answer is!
If they can, can they also do this if the Aggressive Secretary is accessed from Archives? Or from R&D and HQ?
It could possibly be important if a runner has to hit Archives to get a counter to be able to then do an Account Siphon on HQ and use the resulting profit to run on a remote server with an agenda. If the corp can spend the 2C then the runner will not have enough to get into the remote.
Sorry to give you such a weird edge case, but enquiring minds want to know!