Universal distance-hereditary graph

Determine whether there exists a strongly universal countable distance-hereditary graph and, if so, whether one can be chosen to be vertex-transitive.

Background

The paper notes that every finite distance-hereditary graph is a circle graph. This motivates asking whether the class of countable distance-hereditary graphs has a strongly universal member, analogous to the strongly universal countable circle graph established earlier in the paper, and whether such a member can possess vertex-transitivity.