Does the Markov property still hold for time-reverse of a Markov chain?

Question

For a discrete-state Markov chain, the probability of next state is independent of previous states given current state:

$ P(X_{t+1}|X_t,X_{t-1},...,X_1)=P(X_{t+1}|X_t) $

I hope to know whether Markov property still holds for the time-reversal of the above formula. Is the probability of previous state independent of the future states given present state?

$ P(X_{t-1}|X_t,X_{t+1},...,X_T) \stackrel{?}{=} P(X_{t-1}|X_t) $