Tightness of the lower bound under multiple LP solutions

Let the standing assumptions of the paper hold. Let ${\cal S}_{\text{LP}}$ denote the set of solutions to the linear program, and suppose that this set contains multiple solutions. Let $\hat{J}^n(T^n)$ be the diffusion-scaled cost under a policy $T^n$, let $\hat V^n$ be the optimal diffusion-scaled cost, and let $V_0$ be the lower bound from Theorem~. Tightness conjecture. There exist sequences of policies $T^n$ such that

$$\lim_n \hat{J}^n(T^n)=V_0,$$

and consequently

$$\lim_n\hat V^n=V_0.$$

The claim concerns asymptotic optimality in the multiple-solution case, particularly when some optimal modes are degenerate. Earlier results establish tightness for the two-class, two-server case under multiplicity and nondegeneracy, while the conjecture addresses the remaining general case; the supplied text gives no resolution.