Proper containment of interval filament graphs in co-strongly pseudo transitive graphs

An interval filament graph is a graph admitting a representation by curves associated with intervals on the real line, where each curve lies above its interval and connects its endpoints. A graph is co-strongly pseudo transitive of the first type when its non-edges admit an orientation whose associated directed acyclic graph is strongly pseudo transitive of the first type.

Containment conjecture. The class of interval filament graphs is properly contained in the class of co-strongly pseudo transitive graphs of the first type.

The claim asserts both that every interval filament graph belongs to the latter class and that the containment is strict. The supplied excerpt does not indicate whether this statement has been proved or remains open.