The universal property of Baire measurable spaces

Let $\mathsf {BaireMeas}$ be the category of Baire measurable spaces, and let $\mathsf {BaireStoch}$ be its associated Markov category of kernels. A Markov functor is understood to preserve the relevant monoidal and probabilistic structure; a coinflip Markov category is a Markov category equipped with the specified coinflip structure. Baire measurable spaces conjecture. The inclusion

$$\mathsf {BaireMeas} \hookrightarrow \mathsf {BaireStoch}$$

is the initial Markov functor from $\mathsf {BaireMeas}$ into a coinflip Markov category which preserves countable coproducts, pullbacks along coproduct inclusions, and all Kolmogorov products. This would extend the universal-property result for standard Borel spaces to Baire measurable spaces, whose associated stochastic category admits all small Kolmogorov products; establishing the claimed initiality is left as further work.