I hat put the remote into the dock, and it won't work.

Then, last evening, I removed the battery pack from the remote, and put it in again. Then, nothing happened. The remote stayed off.
So the battery packs definitely dead. I tried a battery pack of my dad's walkie-talkie (which had an similiar connector) this morning, and the Pronto went on instantly, and everything worked fine.

So, I ordered a new battery pack from Philips, today. Hope it will arrive soon.