NP-completeness conjecture for APUD(k,m) recognition

For integers $k$ and $m$, let $\mathrm{APUD}(k,m)$ denote the graph class consisting of intersection graphs of disks whose centers lie on $k$ parallel lines, with the relevant representation restricted by parameter $m$.

APUD recognition conjecture. Recognition of $\mathrm{APUD}(k,m)$ is NP-complete.

The paper proves NP-completeness for the special case $m=0$ and asks whether the technique can establish NP membership for general $m$. Thus the general recognition claim remains open in the source.