Why is the martingale factor a martingale in Hansen's 2012 Dynamic Valuation Decomposition? In this question, I'm continuing to explore the tools used/presented in Lars Hansen's Econometrica paper "Dynamic Valuation Decomposition within Stochastic Economies" (2012).

This might be an easy question, but I can't quite see it. In the paper linked above, the factorization is presented in which one component is a martingale.
See p. 937. On this page it presents this formula and says 


  Given a solution to (21), I construct a martingale via
  $$
\widetilde M_t = \exp(-\rho t) M_t \left [ \frac{e(X_t)}{e(X_0)} \right ]
$$
  which is itself a multiplicative functional.


Maybe it's easy, but I just don't see right away how to show that $\widetilde M_t$
is a martingale. How can I show this?

NOTE: This question is related to the following two questions:


Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector
Example of the change of measure proposed in Hansen (2012)
The idea is that after the Perron-Frobenius problem is solved, you have $e$ such that
\begin{align}
E \left[\frac{M_{t+1}}{M_t} e(X_{t+1}) \middle | X_t = x \right]
 &amp;= e(x) \exp(\eta) \\
E \left[\frac{M_{t+1}}{M_t} \frac{e(X_{t+1})}{e(X_t)} \exp(-\eta) \middle | X_t = x \right]
 &amp;= 1 \\
E \left[\frac{\widetilde M_{t+1}}{\widetilde M_t} \middle | \mathcal F_t \right]
 &amp;= 1 \\
E \left[\widetilde M_{t+1} \middle | \mathcal F_t \right]
 &amp;= \widetilde M_t.
\end{align}

Thus we see that $\widetilde M_t$ is a martingale.