Non-universality conjecture for intersection graphs of subdivisions of a fixed graph

Let $Y$ be a fixed graph, and let $\mathcal{G}(Y)$ be the class of intersection graphs of connected subgraphs of subdivisions of $Y$. Non-universality conjecture. For every fixed graph $Y$, the class $\mathcal{G}(Y)$ is non-universal.

This conjecture concerns the range of automorphism groups realizable by intersection-defined graph classes between interval and chordal graphs. The paper presents these classes as an open hierarchy, with the circular-arc case among the motivating examples; the conjecture remains open.