for every $z\in[0,1)$, there is at least one $X^\star(z)\in{\mathbb{R}}$ such that holds. Note that $G$ is constant on intervals where $\omega$ vanishes, so $X^\star(z)$ is not uniquely determined in general. However, it is easily seen that among all the possible solutions $X^\star:[0,1)\to{\mathbb{R}}$, there is precisely one which is monotonically increasing and right continuous, namely $z\mapsto Y(z+(m-1)/2)$. There is an associated $\mu^\star\in{\mathcal{P}}({\mathbb{R}})$ with density in $L^\infty_c$ and corresponding CDF $F^\star:{\mathbb{R}}\to[0,1]$ given by $$F^\star(x) = 
    \begin{cases}
      0 & \text{if $G(x) < (m - 1)/2$},\\
      G(x) - \frac{m - 1}{2} & \text{if $G(x) \in [(m-1)/2, \, 1 + (m-1)/2]$},\\
      1 & \text{if $G(x) > 1 + (m-1)/2$}.
    \end{cases}$$ Similarly as before, we conclude that $\mu^\star$ has the same median as $\omega$, and that the density of $\mu^\star$ coincides with $G'=\omega$ on an interval $[\underline x,\overline x]$ chosen such that $\mu^\star$ has unit mass.

\[thm:q1\] Assume that either $q_a>q_r=1$, or that $q_a=q_r=1$ and $m\ge1$. Let $\mu^\star$ be the unique stationary state determined in Proposition \[prp:steady-stat-equat\] above, and let $\mu:[0,\infty)\to{\mathcal{P}}({\mathbb{R}})$ be a transient solution in the sense of Definition \[dfn:transient\]. Then $W_2(\mu(t), \mu^\star)$ converges to zero monotonically as $t\to\infty$.

Recall that $X(t,\cdot)=X_{\mu(t)}$ denotes the (time-dependent) inverse distribution function of a transient solution. Define $U=\psi_a'\ast\omega$ as 