This would be the adapter, but as Rdsok stated, if it's an eGo battery you shouldn't need it, so don't know what kind of battery you have.
eGo Extender 510 - Outside Threaded for eGo Cone [gotvapes-EGO510extender-adapter] - $3.99 : GotVapes.com, E-cigarette Supplies - Atomizers Cartomizers Mods Juice and more
To be honest, it's a little off-putting to keep having to repeat that I do
have eGo batteries. They came in a box with the eGo logo, and the
manual states the model number is F4306.
If these cartos are designed to work with eGo batteries, then they should work
with eGo batteries. If "posts getting pushed down" are a "feature" of eGo
batteries, then cartos designed specifically for this battery should be designed
accordingly.
Thanks for the link to the adapter, but if, as you say, the cartos should work
with no need of an adapter, then I guess I'd rather wait for specific advice
from the vendor.
If they have nothing more to say than what I've heard so far here, then I
think I will have to return these and leave them to the hobbyist who'd rather
tinker than vape