Approximate Bayesian inversions in probabilistic quasi-Borel spaces

Let $\mathbf{QBS}$ be the category of quasi-Borel spaces, and let $\mathbb{P}:\mathbf{QBS}\to\mathbf{QBS}$ denote the probabilistic construction. For objects $A$ and $B$, a morphism $f:\mathbf{QBS}(A,\mathbb{P}B)$ represents a probabilistic map from $A$ to $B$. Consider an inference functor $F^{\dagger}$ whose action on objects is the identity and whose action on morphisms is similar to $F_q$ above. Approximate Bayesian inversions in probabilistic quasi-Borel spaces. Such a functor should be a dagger endofunctor

$$F^{\dagger}:\mathbb{P}:\mathbf{QBS}\rightarrow\mathbb{P}:\mathbf{QBS}$$

sending objects to themselves and morphisms $f:\mathbf{QBS}(A,\mathbb{P}B)$ to approximate Bayesian inversions

$$f^{\dagger}:\mathbf{QBS}(B,\mathbb{P}A).$$

This would extend the paper's inference construction beyond faithful inverses for string diagrams and provide an approximation to Bayesian inversion in the sense of Cho and Jacobs. Whether the category of quasi-Borel spaces has the conditionals needed to support such a dagger functor is not yet known.