= Solution
Set $a(t)=a_0(t/t_0)^2$. The remaining future <conformal time> is
$$
\eta_\infty-\eta_0=\int_{t_0}^\infty\frac{dt}{a(t)}=\frac{t_0}{a_0}.
$$
A light ray sent to a comoving reflector at separation $\chi$ and immediately returned consumes $2\chi$ of conformal time. The <cosmological round-trip communication horizon> therefore has supremum physical distance
$$
\boxed{d_{\max}=a_0\chi_{\max}=\frac{t_0}{2}=\frac1{H_0}.}
$$
Finite-time reception requires $d_0<d_{\max}$. Indeed the total travel equation is $2d_0/a_0=(t_0/a_0)(1-t_0/t_r)$, giving $t_r=t_0/(1-2d_0/t_0)$. At the limiting distance the return time tends to infinity. This round-trip radius is half the current physical <cosmological event horizon> radius.
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