Objective-value conjecture for the naive semidefinite relaxation of the mKPC

Consider an instance of the multiple knapsack problem with compacity constraints, its linear relaxation $(\mathrm{mKPC})_{\mathrm{LP}}$, and its naive semidefinite relaxation. Write $\operatorname{Opt}(\cdot)$ for the corresponding optimal objective value. Objective-value conjecture.

$$\operatorname{Opt}(\text{naive semidefinite relaxation})\leq \operatorname{Opt}((\mathrm{mKPC})_{\mathrm{LP}})\leq \operatorname{Opt}(\mathrm{mKPC}).$$

The conjecture concerns the ordering of the objective values of the semidefinite relaxation, the linear relaxation, and the original integer formulation. The paper notes that the stronger ROAD property can fail and that feasibility of the ROAD matrix does not ensure semidefinite optimality; the status of this objective-value inequality is not resolved in the supplied text.