= Solution
At fixed mass, decreasing redshift increases the <linear growth factor> and reduces the <halo peak height> $\nu$. During matter domination $\dot D/D=H$ also falls as the universe expands. As long as $\nu>1$, both effects reduce the positive abundance-growth rate, so \b[$t_d$ grows towards lower redshift]. In the very rare-tail limit $t_d\propto D^2/H\propto(1+z)^{-7/2}$ provides the approximate trend.
There is a limit to calling this an increase time. At $\nu=1$, the fixed-mass <Press-Schechter halo mass function> reaches its maximum as a function of time; its logarithmic growth rate vanishes and $t_d$ diverges. For $\nu<1$, the derivative becomes negative, corresponding to net transfer of small objects into more massive systems, so the signed inverse is no longer a positive doubling or increase time. At still later epochs the <cosmological constant> further suppresses the <linear growth factor>. These qualifications prevent extrapolating the positive high-redshift growth trend indefinitely.
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