Martingale conditioned on sigma algebra of stopping time

Question

Let $M_n$ be a martingale in discrete time with respect to the filtration $(\mathcal{F}_n)_{n \in \mathbb{N}}$, and $T$ a stopping time with respect to the same filtration, a.s. bounded by $k$. For $n \geq k$, is $\mathbb{E}[M_n |\mathcal{F}_T] = M_T$?

If not, is something similar true?

Answer 1

Yes, this is an immediate consequence of what is commonly referred to as Doob's Optional Sampling Theorem.