Dynamic Complexity of Parity Exists Queries

Given a graph whose nodes may be coloured red, the parity of number red can easily maintained with first-order update rules in dynamic complexity framework DynFO Patnaik and Immerman. Can this generalised to other or even all queries that are definable logic extended by quantifiers? We consider query asks whether have an edge node is odd. Already simple quantifier structure parity-exists major roadblock for dynamically capturing extensions logic. show cannot quantifier-free rules, variants induce hierarchy such respect arity auxiliary relations. Towards maintaining full it shown degree-restricted maintained.