If AirPod case is dead will AirPods stay on?

I have had this happen twice now. The case is completely drained, but the actual AirPods have an almost full charge. I put them back in the case. The next day I go to use the AirPods they are down to about 2% or dead.

So does the actual case have to have some sort of charge to turn the actual AirPods off?

I assume the case must send some type of electrical signal to the AirPods to tell them to turn off (once they are in the case). However if the case is dead they stay on even while they are in the case. Is this true?

Thanks for your help!