KM7321W/KB700 bluetooth keyboard caps lock and num lock light doesn't reflect status

I have a Dell wireless keyboard (from the KM7321W bundle, which I think is the same as the KB700 standalone keyboard).  It is connected via bluetooth to an Intel AX211 wifi/bluetooth module.  The cap lock and num lock lights don't work right.  When I am actively typing, I can press caps lock or num lock to toggle them, and the on-screen status pop-up shows the status, and when I toggle either to "on" the light is illuminated on the keyboard.  After a few seconds of not typing, though, the keyboard turns off the lights (presumably it goes into a power saving state).  When I push more keys, the keyboard is active again, and the state of caps lock or num lock is not actually changed, but the lights on the keyboard do not reflect the state anymore if either is on.  For instance, num lock may be on, but the light on the num lock key is not on.

I tried to submit a support ticket, but was unsuccessful getting past the "select your product" step (I guess one way to cut down on support costs is to have a website that doesn't let you submit tickets...).  I'm curious if anyone else has this keyboard and can say whether it works properly for them.

I installed the Dell Peripheral Manager SW, and it did some sort of firmware update on the keyboard, but the behavior is the same amount of broken as it was before the update.