[jQuery] superfish navbar issue in Afterburner or JA-Purity Templates

Hello, this is a repost of my first post.

The Superfish navbar menu reverts to a "classic" dropdown menu when in
NAV module of Afterburner template or HORNAV module of JA-Purity
template. It works fine in other module positions and other types of
menu displays behave as they should.

I tried to compare code when the navbar displays correctly in other
module positions and when it doesn't in the NAV (or HORNAV) modules
positions but the CSS is similar. [I've only been with CSS and
Javascript for a few days] and that is why I suspect a bad parameter
handling in a script [without conviction!] that happens only if the
navbar setting is chosen.

I tried on different test sites and the same happens.
I tried disabling JavaScript and the same happens.
MooTools can be on or off with same results.

I'm developping on localhost with FF and the help of FireBug.

I tried so many -small- code changes and settings variations that I am
confused and unable to say if it is CSS or script related. I've got
the unnerving beginner's feeling that it is a simple affair....
Who will show me the way?