If my memory is correct, the pinger (locator beacon) is attached to the memory module rather than the recorder chassis. (Applies to both FDR and CVR of modern design).
It would be a logical design choice, but it would also imply that the pinger power supply batteries are housed in the same module, making it a comparatively heavy item which is more likely to drop straight down to the seabed than, for example, other parts that are mostly aluminium sheet construction.
The BEA must have all this already figured out, of course. Hopefully the memory module and the CVR are soon found.