Cherlin's metric homogeneity classification conjecture

A metrically homogeneous graph is a countable graph whose graph distance is preserved by every isomorphism between finite subgraphs; write $\delta$ for its diameter. The Fraïssé limit of an amalgamation class is the countable ultrahomogeneous structure whose finite substructures are exactly the members of that class. Let $T_{m,n}$ denote Macpherson’s regular tree-like graph with parameters $2\leq m,n\leq\infty$, and let $\mathcal A_3$ and $\mathcal A_H$ be the classes described in the catalogue, with $\mathcal A_3$ 3-constrained and $\mathcal A_H$ of Henson type or antipodal Henson type.

Cherlin's metric homogeneity classification conjecture. The countable metrically homogeneous graphs are the following: those of diameter $\delta\leq 2$, namely the homogeneous graphs classified by Lachlan and Woodrow; in diameter $\delta\geq 3$, the finite graphs classified by Cameron, Macpherson’s regular tree-like graphs $T_{m,n}$ with $2\leq m,n\leq\infty$, and the Fraïssé limits of amalgamation classes of the form $\mathcal A_3\cap\mathcal A_H$ with $\mathcal A_3$ 3-constrained and $\mathcal A_H$ of Henson type or antipodal Henson type.

This is the catalogue attributed to Cherlin for countable metrically homogeneous graphs. The supplied text presents it as a classification and gives no evidence that the classification has been resolved, so its status is left open.