The non-geodesicity conjecture for metric spaces of Reeb graphs

Let $d$ be any metric on the space of isomorphism classes of Reeb graphs. A metric space is geodesic if every pair of points can be joined by a path whose length equals their distance.

Non-geodesicity conjecture. The space of isomorphism classes of Reeb graphs with metric $d$ is not a geodesic space.

This generalizes the preceding conjecture that the intrinsic version $\hat{d}_I$ is not strictly intrinsic. The preceding examples show that the interleaving-distance space and the contour-tree space are not geodesic for $d_I$, but the supplied text gives no resolution of the general claim.