The macaron criterion for reconstructibility of level-2 networks

Let a level-$2$ network be a network whose blobs have level at most $2$, let a macaron be a level-$2$ blob with two cut-edges, and let an alt-path structure be the structure described in the preceding construction. Suppress all macarons and degree-$2$ vertices in the network. Macaron reconstructibility conjecture. A level-$2$ network is reconstructible from its shortest distance matrix if and only if the resulting network does not contain an alt-path structure. The criterion extends the known reconstructibility result from shortest-and-longest distance data to shortest distances alone, while the converse remains unresolved because macaron-added alt-path structures have identical shortest distance matrices.