mc wrote:

I'll check and see if the tube has high voltage across it (apparently not)
and then if high voltage develops with the tube disconnected.

Bear in mind that the typical Geiger tube power supply has a very high
output impedance, and may be loaded down excessively by most common
types of multimeters. If you want an accurate voltage measurement, you
will need something like an electrostatic voltmeter.

Bob Weiss N2IXK