Separating properties $(\kappa_2)$ and $(\kappa)$

Determine whether there exists a separable metrizable space with property $(\kappa_2)$ but without property $(\kappa)$; equivalently, determine whether there exists a separable metrizable space $X$ such that $B_1(X,[0,1])$ is Baire while $B_1(X)$ is meager.

Background

Property $(\kappa_2)$ is defined by requiring that, for every sequence of pairwise disjoint finite sets split into two parts, some subsequence has the two resulting countable unions separated by disjoint $G_\delta$-sets. Property $(\kappa)$ requires every pairwise disjoint sequence of finite sets to have a strongly point-finite subsequence. The paper notes that property $(\kappa)$ implies property $(\kappa_2)$, but does not establish whether the implication can be strict for separable metrizable spaces.

Through the cited characterization of Baire-one function spaces, property $(\kappa_2)$ corresponds to $B_1(X,[0,1])$ being Baire, whereas property $(\kappa)$ corresponds to $B_1(X)$ being Baire. Thus the proposed topological separation is equivalent to asking whether the bounded-valued Baire-one function space can be Baire while the real-valued Baire-one function space is meager.