yes, you are correct that the info boxes don't show up in the next actions view. We have thought about adding that feature, bus so far have not done that.
One thing you can do is quickly navigate to the task in projects and actions view by clicking ALT+Right arrow while on that task to see the checklist items for that task. Then click TAB+2 to quickly go back to the next action view.
Might not be exacatly what you are hoping for, but it is a very quick way to get around this. Thanks!