I wouldn't trust that any further than I could throw it.
If Mugen can't get 3700mAh into a standard size GS4/GS4A battery, ain't nobody else going to be able to do it either, that's my position. Mugen makes a 2750mAh as I noted above and that's pretty much the best we're going to get for the GS4A. That so-called 3700mAh model, nah, I don't believe that capacity for a single moment.
If someone wants to take a chance on it, however, more power to you (pun very much intended).
For us GS4A owners, the Anker or Hyperion dual battery (meaning you get 2 of 'em) and charger setup is probably going to be the best for 'extended' battery life most likely. I'm still considering that myself but I've got 3 batteries already (one Samsung official that's a few weeks old and 2 cheap Chinese counterfeit ones without NFC capability).
Currently in between devices...