I've been looking into this myself. As far as I can discern, the answer is no unless you use M3 lamp assemblies.
Here's a quote from
this thread at post #6: "The M4 LAs don't have a proven rechargeable solution that I know of but you can use M3t LOLA and HOLA with Pila 168s or 17670 from AW in there."
I have a M3 Turbo arriving on Monday and will stick a A19 extender on it and try 2 Pila 168S batteries. I think this will work, tho it may need a JS Burley Pila spring between the LA and the first battery. (Thanks Topper for the info). I'm really not in the mood to spend $4 for 20 minutes of diminishing light with primary 123s, so I hope the rechargable solution works.
I'm on the upside of the learning curve here, so if someone who knows what they are talking about contradicts me, believe them.
I'll advise if I can get the M3T + A19 to make light, assuming my free JSB spring arrives. Thanks JSB!
Edited: I notice in the thread I cited, that Topper is using an M4 with 2 A19s and 3 168S batteries, using the M4 LAs.
best regards