Martingale representation property in progressively enlarged filtrations

Consider $\mathbb{G}$ the progressive enlargement of a filtration $\mathbb{F}$ with a random time $\tau$. Assuming that, in $\mathbb{F}$, the martingale representation property holds, we examine conditions under which the martingale representation property holds also in $\mathbb{G}$. A general methodology is developed in this paper, with results covering every known (classical or recent) examples.