Keyboard navigation in the Build & Run window

Is it possible to navigate the Build & Run tool window with keybard?

My specific use case would be picking a project from the list, opening a context menu and building the project. I have assigned a keyboard shortcut to open the tool window itself, it's just that when it's already focused, the TAB key as well as the arrow keys don't seem to work at all. It's the only thing I have to do with the mouse in my current workflow.

Am I missing something obvious here or should I post it as a feature request?