Metrizable weak $\lambda$-space with non-Choquet real-valued Baire-one function space

Construct a separable metrizable space $X$ such that $B_1(X,[0,1])$ is Choquet but $B_1(X)$ is not Choquet; equivalently, determine whether there exists, at least consistently, a weak $\lambda$-set that is not a $\lambda$-set.

Background

The paper proves that B1(X,[0,1])B_1(X,[0,1]) is Choquet exactly when XX is a weak λ\lambda-space, while B1(X)B_1(X) is Choquet exactly when XX is a Δ1\Delta_1-space. For the separable metrizable setting, the requested example would therefore distinguish the Choquet behavior of bounded-valued and real-valued Baire-one function spaces.

The paper explicitly identifies this function-space question with the set-theoretic/topological question of whether a weak λ\lambda-set that is not a λ\lambda-set can exist, at least consistently. The results established earlier do not resolve the separable metrizable version.

References

Question 2. Is there a separable metrizable space $X$ such that $B_1(X,[0,1])$ is Choquet, but $B_1(X)$ is not Choquet? Note that this question is equivalent to the following (Question 4.6 in ): Is there, at least consistently, a weak $\lambda$-set that is not a $\lambda$-set?

— Some function applications of weak $λ$-spaces  (2608.30278 - Osipov, 31 Aug 2026) in Section 5, Open questions, Question 2

Question 4. Is there, at least consistently, a weak $\lambda$-space which without the property $(\kappa)$? This question is equivalent in a function context to the following: Is there a space $X$ such that $B_1(X,[0,1])$ is Choquet, but $B_1(X)$ is meager?

— Some function applications of weak $λ$-spaces  (2608.30278 - Osipov, 31 Aug 2026) in Section 5, Open questions, Question 4