Polynomial-time recognition conjecture for APUD(1,1)

Let $G$ be a graph, and let $\mathrm{APUD}(1,1)$ denote the corresponding graph class of restricted disk intersection graphs.

APUD(1,1) recognition conjecture. Given a graph $G$, it can be determined whether $G\in\mathrm{APUD}(1,1)$ in polynomial time.

The preceding characterization gives necessary structural conditions for connected graphs in $\mathrm{APUD}(1,1)$, but the source does not establish that they are sufficient. Polynomial-time recognition therefore remains an open problem.