= Solution
For the shallower initial density profile,
$$
M_0(r)=\frac{k'}G r^2,\qquad
M_0=\frac{k'r_0^2}{G}.
$$
Including outer shells, the initial <gravitational potential> is
$$
\Phi_{\rm grav}(r)
=-G\left[\frac{M_0(r)}r+\int_r^{r_0}\frac{\mathrm dM_0(s)}s\right]
=-k'(2r_0-r).
$$
Its central value is finite. With the same remaining-gas escape criterion and fixed-potential approximation as above,
$$
\boxed{r_{\rm quench}(t)
=2r_0-\frac{q}{k'}(e^{\nu t}-1).}
$$
This front is inside the <galaxy> only when $k'r_0\leq Q(t)\leq2k'r_0$. Thus
$$
t_{\rm start}=\nu^{-1}\ln\left(1+\frac{k'r_0}{q}\right),\qquad
t_{\rm all}=\nu^{-1}\ln\left(1+\frac{2k'r_0}{q}\right).
$$
Quenching still moves \b[from the outside inward], but now reaches the centre in a finite time. The contrast with the $r^{-2}$ escape model is the finite central potential, not a reversed gravitational force.
Under the alternative <shell-energy quenching prescription>, the conclusion is different:
$$
B_{\rm shell}(r)=\frac{GM_0(r)}r=k'r,\qquad
\boxed{r_{\rm quench}^{\rm shell}(t)=\frac q{k'}(e^{\nu t}-1).}
$$
Here the quenched region is inside the front and grows \b[from the centre outward], unlike the simultaneous quenching of the isothermal shell-allocation model. Its edge reaches $r_0$ at $\nu^{-1}\ln(1+k'r_0/q)$. This change of direction is a consequence of the chosen local energy model: $GM(r)/r$ increases with radius for this profile, whereas the true binding potential including all outer shells always becomes shallower outward. The source does not identify which local allocation it intends; keeping the conventions explicit resolves the different conditional predictions without discarding the exterior-shell potential.
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