The generic constancy conjecture for causal stochastic solutions

Let $\Pc_{\mathrm{in}}^-$ be the space of input probability laws and $\Pc_{\mathrm{state}}^-$ the space of state–input probability laws, with projection

$$(\pi_{\US^-})_* \colon \Pc_{\mathrm{state}}^- \rightarrow \Pc_{\mathrm{in}}^-.$$

Let $\#\Sc^{\mathrm{c,stoch}}(\Xi)$ and $\#\Oc^{\mathrm{c,stoch}}(\Xi)$ denote, respectively, the numbers of causal stochastic solutions and causal stochastic outputs associated with an input law $\Xi$. Assume that the state-space system measurably distinguishes reachable states when the relevant reachable-state relation is Borel measurable and admits a measurable readout of the next output.

Generic constancy conjecture. Suppose that $\Pc_{\mathrm{in}}^-$ and $\Pc_{\mathrm{state}}^-$ are Polish and that $(\pi_{\US^-})_* \colon \Pc_{\mathrm{state}}^- \rightarrow \Pc_{\mathrm{in}}^-$ is proper. Then, $\#\Sc^{\mathrm{c,stoch}}(\Xi)$ is generically constant, and so is $\#\Oc^{\mathrm{c,stoch}}(\Xi)$ if the state-space system measurably distinguishes reachable states.

This is posed as an open problem because the paper does not know whether the causal stochastic FMP holds generically; the conjecture would yield generic control of bifurcations of causal stochastic solutions and outputs.