Before revision:  We prove existence of solutions to the corresponding conditional McKean--Vlasov equation  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 system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.