= Solution
Use the same affine drift cancellation with terminal conditions $a(0)=0$, $b(0)=1$. It gives $a(\tau)=0$, $b(\tau)=e^{-\tau}$, so
$$
D_te^{-(T-t)}r_t
=\mathbb E^{\mathbb Q}[D_Tr_T\mid\mathcal F_t].
$$
The process is bounded, which justifies the <conditional expectation> identity. The <Bayes formula for conditional expectation> for the <forward measure> now gives
$$
\begin{aligned}
\mathbb E^{\mathbb Q_T}[r_T\mid\mathcal F_t]
&=\frac{\mathbb E^{\mathbb Q}[D_Tr_T\mid\mathcal F_t]}
{\mathbb E^{\mathbb Q}[D_T\mid\mathcal F_t]}\\
&=\boxed{\frac{e^{-(T-t)}r_t}{1-(1-e^{-(T-t)})r_t}}.
\end{aligned}
$$
The denominator is positive; the ratio lies in $[0,1]$ and equals $r_T$ at maturity. This is the <forward-measure terminal rate in a linear bond model>.
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