The degree-nine conjecture for rational plane curves with triple-point singularities

A reduced plane curve has multiplicity three singularities when all its singularities have multiplicity $3$. Degree-nine conjecture. There is no reduced plane curve in any characteristic of degree more than $9$ whose components are all rational and whose singularities all have multiplicity $3$.

This is proposed as a more accessible target related to the multiplicity-sequence conjectures. The source supplies no resolution, so it remains open.