No I don't find a new N-representability condition.

Since 1980, in my opinion, no essential progress in N-representability condition other than Braams' JCP 2004 T1/T2 paper. Even, Braams's conditions are originally from Erdahl's 1978 paper. Trivial one is Erdahl and Jin's k-th order approximation (Mazziotti calls k-positivity). I'm very sure we all know them. Erdahl implicitly uses it, of course. otherwise how Braams extract from Erdahl's paper? :-)
I's a pity I couldn't find any applicable new N-representability conditions.