Monolithic spaces of measures

For a compact space K, we consider the space P(K) of regular probability measures on K, equipped with the weak* topology. For zerodimensional spaces K, we can, equvalently, speak of  P(A), the space of finitely additive measures on a Boolean algebra A (with the topology of converence on elements of A).
A compact space X is $\omega$-monolithic, if every separable subspace of X is metrizable. We shall discuss possible characterizations of those spaces K for which the space P(K) is $\omega$-monolithic.