Can I update iOS on a locked phone? Tim Cook's open letter about the FBI's attempts to access a locked iPhone 5c have me wondering a fairly simple question that I haven't found an answer for:

Assumptions:


I have an encrypted iPhone
It is set to wipe after 10 failed passcode attempts
It is currently locked
I only use Apple-approved methods


Question:

Can I update iOS on the device without unlocking it and without losing/wiping  the data?

Update:

As an aside to the actual question, I see that the FBI is not trying to do what I'm asking. They are asking for a version of iOS that they would load directly into RAM and run from RAM without making any modifications to either the OS partition or the user partition.
You cannot update iOS on a locked device. When you try to update via iTunes(which is the only possible in this situation), it will ask you to unlock the iPhone. It is simply not possible to update or restore a locked iPhone or any iOS device.