Let $X$ be a non-empty finite set, let $\mathcal{M}(X)$ denote the set of metrics on $X$ for which if $\rho\in \mathcal{M}(X)$ then $\rho(x,y)\ge 1$ for all $x,y\in X,\, x\neq y$ (i.e. distinct points must be at a distance of at-least $1$) which we equip with the Gromov-Hausdorff distance $d_{GH}$, and let $\mathcal{P}(X)$ denote the set of probability measures on $(X,2^X)$. Now, for any given $\rho\in \mathcal{M}(X)$, we denote the $1$-Wassertein distance with respect to the metric structure $(X,\rho)$ by $$ \mathcal{W}_\rho(\mu,\nu) = \inf_{\pi\in \mathcal{P}(X^2);p^1_{\#}\pi = 1,\, p^2_{\#}\pi=\nu} \int \rho(x,y) d\pi(x,y) $$ where $\mu,\nu\in \mathcal{P}(X)$ and $p_i$ are the canonical projections of $X\times X$ onto the $i^{th}$ coordinate for $i=1,2$.
Is there a name for this distance between probability measures: $\mu,\nu \in \mathcal{P}(X)$ $$ \inf_{d\in\mathcal{M}(X)}\, \mathcal{W}_{\rho}(\mu,\nu) + \operatorname{dim}(X,d) \tag{1} $$ where $\operatorname{dim}(X,d)$ is the doubling dimension of $(X,d)$; i.e. the log doubling constant.
Note the doubling dimension of $(X,I_{n\neq y})$ is that of the complete graph which is $\log_2(|X|)$ which is maximal; thus, as noted by Martin Hairer in the comments, without this penalty the value of problem (1) is $TV(\mu,\nu)/2$.
Is this object studied in the literature? How can one compute/estimate the optimal value of (1)?
