Forward-convex convergence in probability of sequences of nonnegative random variables

For a sequence $ (f_n)_{n \in \mathbb{N}}$ of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence $ (h_n)_{n \in \mathbb{N}}$ with $ h_n\in \mathrm {conv}(\{f_n,f_{n+1},\dots \})$ for all $ n \in \mathbb{N}$ to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.