[fmos]

I have tested taskjuggler, but it fails to schedule tasks when a mixture of with/without deadline (end) tasks is provided.

The only software I am aware of that seems fitting is SkedPal https://www.skedpal.com/

I was so surprised that this isn't easier to find that I posted it on HN just now.

Unfortunately though (for me) there does not seem to be a (paid) self-hosted (privacy aware) version available.