Try powering it on (and turning on the source), then simply walk away for 10-15 minutes?
My BV10-46 requires 12 minutes powered up before the sound finally cuts in.  After 2++ hours, it cuts out again!  Because of the out-of-warranty expense (incl. 💸travel fees, USA) I never called for service.  Whether the temperature is affecting a chip solder pad or just a poorly seated ribbon cable, I'll never know.

this didn´t really help ;-)

Oh well; long shot.  (My PowerLinks light up as you describe, and also on the remote List/Speakers/1, 2, 3 turn them and the main speaker on & off, but no sound comes thru on any group.  (Until it finally does, in my case not yours.))

A ***possible*** work-around -- though that is not what you are looking for -- is: BV10 Headphones Out jack is in the main connection panel, beside the camcorder input jacks.  If you mute the speakers with the remote (via List/Mute or the middle of the volume up/down button) then supposedly the volume control now controls the headphones volume. That plus a 3.5mm->2xRCA should do the trick to cable to your Beolab 4000s, which have line-in audio sensing for their own power-up.  (Set their toggles for Line-In/Left/Right to the single RCA jack.) That might work ***if*** the headphone jack is driven separately from the main speakers?

I am doubtful that this works, but I never actually tried to listen on the headphone jack while my speakers were broken-muted, so maybe?  Either way, it's a debugging data point if you call for service or go rooting around inside yourself.  I'm curious how you finally resolve the issue and hope you report here!

[Edit: Turns out BV10s can tell by impedance whether headphones or some other device is connected!  So Speakers Mute is unnecessary because it would mute the (realized-to-not-be) headphones too.  Too damn smart.  Alas, this behavior also implies that the headphones jack and the speaker output are separated too late in the chain for the above workaround.  But at least it's easy to test...]

That´s quite a bit more information.

Unfortunately the headphone jack is also dead.

My current plan is: connect a fire tv stick into a HDMI Audio extractor. Video through HDMI to the BV10. Use the speaker output of the extractor  to connect them to the powerlink sockets (left & right). in parallel input the speaker wires into a digital amplifier which will be connected through a frequency crossover to the speakers in the BV10.

so, small update as the hdmi extractor arrive. I have sound :-)
but, the beolab are connected through chinch and the extractor has a only a physical volume control - i wisely ordered one with, assuming the hdmi has a full signal. It would have been soooo loud.

Now i am going to add a passive amplifier (found an old denon avr reciver in my cellar). HDMI etractor via optical into the denon, from there to the Beolabs and the internal speakers of the beovision. With temote control, which will be exchanged by a nice all in one, where i can control the beovision, the fire tv stick and the receiver.

 

Not as nice as the original - but it works and i really can´t throw away a beovision 10-40 in (picture) working condition...

Now i am awaiting some spare pcbs from a friend in slovenia, maybe i can exchange my internals with them and we are back on the original train. In the meantime - i learned a lot about the beovision.
All the best
69er