Terminality of the Radon monad in the category of endofunctors of probability measures

Determine whether the pair consisting of the Radon monad R on KHaus and the restriction isomorphism res: H R → G H is terminal in the comma category (H_* ↓ G H), whose objects are endofunctors S on KHaus equipped with natural transformations β: H S → G H, where H: KHaus → Meas assigns the Baire σ-algebra and G is the Giry monad on Meas. Concretely, decide whether for every (S, β) there exists a unique morphism (S, β) → (R, res) in (H_* ↓ G H).

Background

Section 4 studies a universal property of probability monads as terminal objects in categories of endofunctors of probability measures. For KHaus with the Baire functor H, Corollary A.4 establishes an isomorphism res: H R → G H relating the Radon monad R to the Giry monad G composed with H.

The example raises whether (R, res) is terminal in (H_* ↓ G H). Proposition 4.9 and Example 4.10 subsequently show terminality within a full subcategory determined by continuity of the component β[0,1], but they do not settle terminality in the entire (H* ↓ G H).