Characterization of local minima of the $1$-Wasserstein population functional

Let $X_1,\ldots,X_m$ be persistence diagrams with only finitely many off-diagonal points. For a persistence diagram $Y$, define

$$F_1(Y)=\frac{1}{m}\sum_{i=1}^m d_1(X_i,Y).$$

For each optimal pairing from $W=\{w_j\}$ to $X_i$, write the image of $w_j$ under the pairing as $\phi_i(w_j)$, and call a point the median of a finite collection of points when its coordinates are coordinatewise medians. Local-minimum median characterization. $W$ is a local minimum of $F_1$ if, for any set of optimal pairings from $W$ to each of the $X_i$, denoted by $\phi_i$, every $w_j$ is the median of

$$\{\phi_i(w_j)\}_{i=1,2,\ldots,m}.$$

This gives a necessary characterization of local minima for the median functional on persistence diagrams, extending the analogous one-dimensional median condition. The source provides no resolution status beyond presenting the statement in a conjecture environment.