question about weak omega category

[Philippe Gaucher <gaucher@irmast1.u-strasbg.fr>]

Bonjour,


Let us call cubical omega-category a cubical complex
with connections and operations +_j like in the
paper "On the algebra of cube", Brown & Higgins or
like in Al-Agl's PhD "Aspect of multiple categories".
There is a conjecture which claims that the category 
of cubical omega-categories is equivalent to the category 
of globular omega-categories. If I understand correctly, 
the conjecture was proved  in some richer framework 
but seems to be (in my knowledge) still open as stated 
above. 

My question is : is there a similar conjecture for 
weak omega-category ? Is there a notion of cubical
weak omega-category somewhere in the literature
and a notion of globular weak omega-category ?

Any reference is welcome. I have found nothing with
the usual research engine but maybe I did not use
the good key-word.


pg.