The hyperimaginary elimination conjecture for Urysohn monoids

Let $\mathcal{R}$ be a nontrivial Urysohn monoid. Define

$$\operatorname{heq}(\mathcal{R})=\{\alpha\in\operatorname{eq}(\mathcal{R}^*):\alpha<\inf\{r\in\operatorname{eq}(\mathcal{R}):\alpha\leq r\}\}.$$

Hyperimaginary elimination conjecture. The theory $\operatorname{Th}(\mathcal{U}_{\mathcal{R}})$ has elimination of hyperimaginaries if and only if $\operatorname{heq}(\mathcal{R})=\emptyset$.

The conjecture would characterize elimination of hyperimaginaries for theories of generalized Urysohn spaces. Combined with the proposition that a countable distance monoid of finite archimedean rank has empty $\operatorname{heq}(\mathcal{R})$, it would imply that finite strong order rank for $\operatorname{Th}(\mathcal{U}_{\mathcal{R}})$ entails elimination of hyperimaginaries.