Hi Nicolas.
I've already made that check and various combinations too, trying to figuring out the problem, so I've noted that as you select the switcher it clears the previous settings.
Look at the attachments: the problem is only related to the registration page, while on the cart everyhting seems to work perfectly.
So I've checked the menu item and the Hika Registration plugin; the item ID in the registration plugin is correct.
Looking at the codes there's no display:none properties that could hide the password fields.
As said before, there's no hurry. Have a good weekend.
Davide