 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 $t_d$ 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.