The forbidden-isometric-subgraph characterization of half-hyperbolic 6-chordal graphs

Let a graph be 6-chordal if every cycle of length at least six has a chord, and let graph hyperbolicity be measured in the usual graph metric. Let $G_1,G_2,G_3,C_4,C_6,H_i$ for $i=1,\ldots,5$ be the ten special graphs depicted or defined in the paper. An isometric subgraph is a subgraph whose induced distances agree with the distances in the ambient graph.

Forbidden-subgraph conjecture. A $6$-chordal graph is $\frac{1}{2}$-hyperbolic if and only if it does not contain any of the ten special graphs $G_1,G_2,G_3,C_4,C_6,H_i$, $i=1,\ldots,5$, as an isometric subgraph.

The claim gives a finite forbidden-isometric-subgraph characterization of half-hyperbolic graphs within the class of 6-chordal graphs. The supplied source context does not indicate whether this conjecture has been proved or disproved.