Direct-observation conjecture for filter stability

Let $(X_k,Y_k)_{k\,in\mathbb{Z}}$ be a stationary infinite-dimensional hidden Markov model as in Section 3, with $X_k\in\{-1,1\}^{\mathbb{Z}}$ and $Y_k\in\{-1,1\}^{\mathbb{Z}}$. Suppose

$$Y_k^v=X_k^v\xi_k^v,$$

where $(\xi_k^v)_{k,v\in\mathbb{Z}}$ are independent and identically distributed, independent of $X$, and satisfy $\mathbf{P}[\xi_k^v=-1]=p$. Direct-observation filter-stability conjecture. If the underlying process $(X_k)_{k\in\mathbb{Z}}$ is stable, then the filter is stable. The direct observation structure has no observation symmetry when $p\ne\frac12$, since distinct configurations induce distinct observation laws; the case $p=\frac12$ is trivial because the observations are independent of the signal. Whether stability is always inherited in this setting remains open.