Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ such that $W_1(\mu,\lambda_\alpha) \le \alpha W_1(\mu,\nu)$ and $W_1(\nu,\lambda_\alpha) \le (1 - \alpha) W_1(\mu,\nu)$ ?

Observations

Thus $\lambda_\alpha$ is some sort of interpolation between $\mu$ and $\nu$. Maybe the Wasserstein barycenter of $\mu$ and $\nu$ (with weights $\alpha$ and $1-\alpha$) or the Mccann interpolation can do the job ?