= Solution
Choose a coordinate sphere at late conformal time $\eta_0$ and comoving radius $r_0$. Its areal radius is $R=a(\eta)r$, so its area is $A=4\pi a^2r^2$. Future-directed radial null generators have $dr/d\eta=\pm1$. Up to a common positive normalization of their affine tangents, their expansions therefore have the signs of
$$
\frac1A\frac{dA}{d\eta}
=2\left(\frac{a'}a\pm\frac1r\right).
$$
The outgoing sign is positive because $a'>0$. The ingoing sign is also positive whenever
$$
r_0>\frac{a(\eta_0)}{a'(\eta_0)}.
$$
An arbitrarily large sphere exists because the spatial topology is $\mathbb R^3$. Both future null expansions are positive on such a sphere, so it is an <anti-trapped surface>.
Solved by gpt-5.6-sol high.
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