Matt Whiting wrote:
Every VOR I've flown has a needle the swings wildly from side to side or
goes full-scale to one side when crossing a VOR. Doesn't matter how
precisely the VOR is crossed. The off flag appears as well, but the
needle doesn't just sit in the donut through all of this. Maybe you'll
believe the Air Force.

http://www.vusaf.org/aetc/documents/VOR%20MANEUVERS%20_PART%202.pdf#search='vor%20cone %20of%20confusion'


From the referenced document:

The cone of confusion is an area above the station where the signal
from the antenna is not receivable and therefore is not reliable.

So, yes, I do believe the Air Force. Show me a CDI that does not center
the needle when the VOR signal is lost.

I also challenge your claim that the needle will swing "wildly from side
to side." Passing slightly abeam the station will cause the needle to
go full deflection in one direction and stay there until after station
passage. In a properly functioning CDI, there should be no side-to-side
swinging.