Ive got a single cell (18650) LED lenser power bank and it will charge my cellphone with the lenser batt but not with an Ali Express special . However in the headlamp MH10 single cell (18650) the Ali batt works fine snd costs about 1/4 as much. I think its just the power-bank application is very demanding as one Li Ion batt doesnt really have the 5V voltage and must need to be magically stepped up ( contradicting the principles of DC electroteknik.)
Check the mAmp.hr ratings.
If the headlamps themselves are old I personally would buy two cheap ones so I know ive always got one fully charged batt on me.
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