Alpha detection is not required and so as to protect the mica window.
Most tubes are filled with a mix of inert gas and a halogen gas... the
latter can sometimes attack the seal around the mica. So, if your GM
tube has a mica window, that would be the typical failure mode. If
not, the wire-lead seals sometimes fail.

http://www.imagesco.com/articles/geiger/01.html

Some basic information here.

Thanks. I don't think this one has a mica window. I'll check. The circuit
board has had wax poured on it (by the maker, for insulation).

When you say "the wire-lead seals sometimes fail" do you mean they let air
leak into the tube, or they fail in some other way that might be fixable?