Induced disjoint paths in 1-string and segment intersection graphs

Determine the computational complexity of Induced k-Disjoint Paths in 1-string graphs and in segment intersection graphs.

Background

The hardness construction in the paper produces string graphs, but the authors do not obtain representations in which every pair of strings intersects at most once. This leaves open whether the hardness persists in the narrower class of 1-string graphs.

Segment intersection graphs form a further restriction of 1-string graphs, where the representing strings are line segments. The authors explicitly include this class as a second unresolved setting.