Can you provide any proof of this? What will id be working on?

Nope, it's solely hearsay, even to me.
Look at it this way, though - id has fallen way behind. id tech 4 was used for only two non-id projects, one of which is by Splash Damage, who had a ton of experience with the engine from Quake Wars and Wolfenstein, so it hardly counts.
id hasn't put major effort into developing easy-to-use tools, and it's a monumentally uphill battle these days. An entire engine cottage industry has popped up of engines as powerful as their own (at least on consoles) with easier to use tools that have enormous amounts of tutorials out there. They always created their tools more for their own use than for others to easily learn on.

id may have damn near created the engine licensing subset, but they never focused on it. As a result they've kind of been edged out of it. Whether they're interested in reversing that or not is not something I can answer with authority, but I've been told ZeniMax recognizes the problem and realizes the solution is too costly.