If the moving average of a process is a martingale, is the process a martingale?

Problem set up:

Let $\mathcal F_t$ be a filtration satisfying the usual conditions. Let $T > 0$ be a fixed real number, and define the filtration $\mathcal H_t := \mathcal F_{T + t}$.

Suppose a cadlag adapted process $X$ is almost surely locally integrable, i.e. for any compact set $C \subset \mathbb R_+$we have $\int_C X_t \ dt < \infty$ a.s.

Define the moving average process $M$ by $M_t := \int_{[t, t + T]} X_s ds$.

Suppose that the following conditions hold:

• Almost surely, $X_0 = X_s$ for all $s \leq T$.

• $M_t$ is a $\mathcal H_t$-martingale.

Question: Is it true that $X$ is an $\mathcal F_t$-martingale?