Is conditional expectation same for all periods for stationary process?

Question

Suppose there is a stationary process Y(t)

I know that E[Y(t)]=E[Y(t-k)] must hold for any k.

But does

E[Y(t)|Y(t-1)] = E[Y(t-1)|Y(t-2)]

necessarily hold?

Not sure if the conditional distribution, so that conditional expectation, is the same across time.