$ E[Y| {F}_{\leq t}]= E[Y]$ for Markov process

Question

Let $ X_s, ~ s\in [0,\infty) $ be a real valued stochastic process with the Markov property. That is

$$ E[Y| {F}_{\leq t}]=E[Y| {F}_{= t}]$$

Where $ Y $ is $ {F}_{\geq t} $-measurable, and ${F}_{\leq t} ,{F}_{= t}, {F}_{\geq t} $ are the sigma algebras generated by the respective sets of $X_s$.

Now if $ Y $ is $ {F}_{\geq u} $-measurable, for some $ u > t $, then it seems like we should have

$$ E[Y| {F}_{\leq t}]= E[Y]$$. Is that true and how to prove?

Answer 1

Counter-example: take $X_t=tX, Y=X_u=uX$ where $X$ is a fixed non-constant random variable.