Kunita's exchange conjecture for purely nondeterministic hidden Markov models

Let $(X_k,Y_k)_{k\in\mathbb{Z}}$ be a stationary hidden Markov model with hidden state space $E$ and observation space $F$. The hidden process is purely nondeterministic if

$$\mathbf{E}\big[|\mathbf{P}[X_n\in A|X_0]-\mu(A) |\big] \xrightarrow{n\to\infty}0$$

for every $A\in\mathcal{B}(E)$, where $\mu$ is the invariant distribution of $X_0$. The observations are nondegenerate if there exist a $\sigma$-finite reference measure $\varphi$ on $F$ and a strictly positive measurable function $g:E\times F\to]0,\infty[$ such that

$$\Phi(x,A) = \int \mathbf{1}_A(y)\\,g(x,y)\\,\varphi(dy)$$

for all $x\in E$ and $A\in\mathcal{B}(F)$. The exchange identity is the equality between the intersection and supremum of the relevant sigma-fields, denoted by

$$in the source. **Kunita's exchange conjecture.** If the hidden process is purely nondeterministic and the observations are nondegenerate, then$$

holds true.

This conjecture would extend the known result from absolutely regular hidden processes to purely nondeterministic ones. The source states that it was conjectured in the cited work and provides no resolution here.