ZFC separation of properties $(\kappa_2)$ or $(\kappa)$ from weak $\lambda$-spaces

Construct in ZFC a space $X$ with property $(\kappa_2)$ or property $(\kappa)$ that is not a weak $\lambda$-space.

Background

The paper establishes that every weak $\lambda$-space has property $(\kappa_2)$, and it also recalls that property $(\kappa)$ implies property $(\kappa_2)$. Under Martin's Axiom, it gives a space with property $(\kappa)$ that is not a weak $\lambda$-set, but this does not provide a ZFC example and concerns the set version of the terminology.

Question 3 asks whether the separation can be realized in ZFC: specifically, whether a space satisfying the stronger combinatorial property $(\kappa)$, or at least property $(\kappa_2)$, can fail to be a weak $\lambda$-space.