Ramsey-cofinite versus Ramsey-lower-dense conjecture

Let $G$ be a graph. A graph is Ramsey-cofinite when it has the cofinite monochromatic embedding property defined in the paper, and it is Ramsey-lower-dense when it has positive Ramsey lower density in the paper's terminology. Ramsey-cofinite conjecture. If $G$ is not Ramsey-cofinite, then $G$ is not Ramsey-lower-dense. The source says this is suspected for every graph and relates it to a weaker previously asked question; no resolution is given.