Before revision:  We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional  system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.
After revision:  We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.