= Solution
Keep the maturity $T$ fixed and write $F(t,T)=\exp L(t,x_t)$, where
$$
L(t,x)=ax+(1-a)\widetilde\mu+\frac{\sigma^2}{4\alpha}(1-a^2),\qquad a=e^{-\alpha(T-t)}.
$$
As calendar time increases, $\partial_ta=\alpha a$, so $L_x=a$, $L_{xx}=0$ and $L_t=\alpha a(x-\widetilde\mu)-\sigma^2a^2/2$. Applying the <Itô formula> to $L$ and then to its exponential gives
$$
dL=-\frac12\sigma^2a^2dt+\sigma a\,dZ_t,\qquad \boxed{\frac{dF(t,T)}{F(t,T)}=\sigma e^{-\alpha(T-t)}\,dZ_t}.
$$
The <Itô formula>'s exponential correction cancels the displayed drift of $L$. The zero drift also follows from the <tower property of conditional expectation>: for $t\leq u\leq T$, $\mathbb E_t[F(u,T)]=\mathbb E_t[S_T]=F(t,T)$. In particular $F(T,T)=S_T$, while the quote's instantaneous <spot volatility> approaches $\sigma$ as expiry approaches.
This is a <martingale> under the same measure used in the stipulated <conditional expectation>. To interpret it as <futures pricing>, that measure must be a suitable pricing measure. For a non-dividend-paying traded <stock> at a deterministic <interest rate>, the absence of <arbitrage> instead gives a <forward contract> delivery price $S_t e^{r(T-t)}$. A physical-measure expectation under the given mean-reverting drift is not automatically that price; the question's expectation convention is sufficient for the calculation above, but not for this additional identification.
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