I guess I did not think about airguns requiring suppressors. But, following up on your comments, I found this discussion on pyramydair forums:

"SO WHAT? We're talking about airguns here!
Yes we are, and most airgunners know that the Bureau of Alcohol, Tobacco, Firearms and Explosives (BATFE) has jurisdiction over firearms ONLY. In fact, U.S. Code specifically prohibits any federal, state or local municipality from declaring an airgun to be a firearm. Not only do the federal firearms laws NOT apply to airguns, they CANNOT be applied by law! But that doesn't get us out of the woods.

However...
IF you have an airgun silencer, and IF it can be removed and installed on a firearm, and IF it then quiets that firearm - that silencer meets the legal definition and is subject to the law. However, in the United States, a person is considered innocent until proven guilty, so it is also necessary to prove intent, as in you intended to use the silencer in your possession on a firearm."

I will be interested to see what you learn from the game wardens where you live.