Cron: pages only available on one lang on source are visible on destinations (only one another lang on destination)

I imported the online demo + validated all the pages in FR + EN. Then I made a cron on a second site which only exist in FR.

If I then create some pages only visible in EN on my source (e.g. in my left menus), I see some N.D. pages on my destination site. It looks like working for the top nav bar. Are the checks in the templates different for the top menu and the left menus? Plz check and address this issue with Joe if required.