Has anyone experimented with installing some sound deadening sponge or foam inside the BK9 to minimise the Keybed noise?

I have pulled apart my BK9 and it is quite hollow inside so I feel that the casing of the instrument itself Is actually one of the main reasons why the Keybed is so loud, especially when you release the keys instead of actually pressing them...

The BK9 Keybed feels beautiful to play but it is one of the loudest I have ever come across...