I was hopeful that the issue was a bad battery (as these are relatively easy to solve), so I decided to start with the charger-side earpiece.

There are subtle holes on the outside rings of either earpiece, hinting that prying may be possible. The surrounding plastic is very fragile, so if you are trying to salvage the suave exterior, you may want to apply heat as there is strong adhesive holding the ring in place.

Be careful if heating - the plastic will likely warp with higher temperatures (try to keep just warm/medium hot to the touch), and use broad, plastic prying tools for best outcome.

After removing the plastic ring, screws were now exposed that held two halves of the outer earpiece assembly together. Excellent!

Though the screws provided support, the two halves didn't simply fall apart - a surrounding plastic clip holds the two pieces snugly together. Time to grab the spudger...