Universally meager sets and the Baire category property of some function spaces

We shall discuss the problem of inner characterization of topological spaces $X$ for which the space $B_1(X)$ of real-valued Baire class one functions is Baire or Choquet. We prove that for any separable metrizable space $X$ the following implications hold: ($X$ is a $\lambda$-space)<=>($B_1(X)$ is Choquet)=>($B_1(X)$ is Baire)=>($X$ is universally meager). We do not know if the universal meagerness of $X$ is equivalent to the Baireness of the function space $B_1(X)$.