Given a probability space $(\Omega, \mathcal F , \mathbb P)$, a filtration $\mathbb F = (\mathcal F _t )_{t\geq 0}$ and $\mathbb F$-adapted brownian motion $W=(W_t)_{t \geq 0}$, consider $X^{t,x}= (X_s ^{t,x})_{s\in [t,T]}$ the (markovian) solution of the following SDE
$$ X_s^{t,x}= x + \int _s ^T b(s,X_u ^{t,x}) du + \int _s ^T \sigma(s,X_u ^{t,x}) dW_u $$ under the standard assumptions on $b, \sigma $ for existence and uniqueness of the strong solution.
Let $f: [t, T] \times \mathbb R \longrightarrow \mathbb R $ a bounded function. Can we say that $\mathbb E[\int _{t_1} ^{t_2} f(s, X_s^{t,x} )ds \mid \mathcal F _{t_1} ]=0$ (with $t\leq t_1 \leq t_2 \leq T$)? If yes, how to show it ?