Equality conjecture for the PRs cubed and PRS relaxations

Let $\mathcal{S}^\kappa_{P,R,s^3}$ and $\mathcal{S}^\kappa_{P,R,S}$ denote the relaxations associated respectively with the PRs$^3$ and PRS formulations. Equality conjecture.

$$\mathcal{S}^\kappa_{P,R,s^3}=\mathcal{S}^\kappa_{P,R,S}.$$

The conjecture asserts that the two relaxations have exactly the same strength. The supplied text gives no resolution, so its status remains open.