Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying $$ P_1(X,d)\triangleq \left\{ \nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\nu(x) <\infty \right\}, $$ which will be equipped with the 1-Wasserstein metric $d_{W_1}$.

In Section 3.1 of this text, it is remarked that the existence of a $1$-Lipschitz map $\beta:P_1(X,d)\rightarrow (X,d)$ satisfying $$ \beta(\delta_x)=x, $$ implies that $(X,d)$ is a Busemann space; hence does not exist in general.

**Question:**
If the requirement the $\beta$ is $1$-Lipschitz is relaxed to the existence of a lsc function $\rho:(0,\infty)\rightarrow (0,\infty)$ satisfying $\lim\limits_{x \downarrow \infty} \rho(0)=0$ and
$$
\rho(d_{W_1}(\beta(x),\beta(y)))\leq d(x,y),
$$
then does this still imply the existence of geodesics on $(X,d)$. In particular, do such maps exist in general for some such $\rho$?
For example, $\rho$ may be
$$
\rho(x)\triangleq L x^{\alpha},
$$ for some $\alpha \in (0,1)$ and $K>0$.