Wasserstein (Earth movers) distance between two Beta distributions

Question

Consider the Beta distribution with density $$ p(x; \alpha, \beta) = \frac{1}{B(\alpha,\beta)} x^{\alpha-1} (1-x)^{\beta-1} $$ where $B$ is the normalizing Beta function and $\alpha,\beta$ its shape parameters. I am looking for the $W_1$ Wasserstein metric and associated transport plan between two shapes $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$, assuming that the cost of a move is simply the distance between the two points, $c(x,y) = |x-y|$.