Positive Ramsey upper density conjecture for the graphs $H_d$

For each $d\in\mathbb{N}$, let $H_d$ be the graph defined in the source's Proposition referenced by prop:constr, and let $\operatorname{\overline{Rd}}(H_d)$ denote its two-color Ramsey upper density. $H_d$ density conjecture. There exists $c=c(d)>0$ such that

$$\operatorname{\overline{Rd}}(H_d)\geq c;$$

more weakly, $\operatorname{\overline{Rd}}(H_d)>0$. The conjecture is proposed to imply the cited question for graphs without a finite dominating set; the supplied text gives no resolution.