The conditional Poincaré inequality inheritance conjecture for filters

Let $X$ be a Markov process satisfying the standard Poincaré inequality with constant $c_0$, and let the filtration ${\cal Z}$ be obtained using the observation model. The conditional Poincaré inequality is

$$\operatorname{enr}^{\bar{\mu}}_T(F)\geq c\,\operatorname{var}^{\bar{\mu}}_T(F)\quad\forall F\in L^2_{{\cal Z}_T}(\mathbb{S}),\;\forall T\geq0.$$

Conditional Poincaré inequality inheritance conjecture. Suppose the standard form of the Poincaré inequality holds for $X$ with constant $c_0$, and suppose ${\cal Z}$ is obtained using the observation model. Then the conditional Poincaré inequality holds with a positive constant $c$. This conjecture asserts that filter observations preserve enough of the underlying Markov process's Poincaré structure to yield a positive conditional constant. The supplied text gives no resolution or general conditions guaranteeing the claim.