Given a graph $G$, it is known independence number $\alpha(G)$ and Lovasz theta $\vartheta(G)$ satisfy the inequality, $\alpha(G^{\boxtimes n}) \leq \vartheta(G)^{n}$.
If $\alpha(G^{\boxtimes n}) < \vartheta(G)^{n}$, under plausible complexity theory conjectures, is it possible to have any other polynomial time computable function $\psi(G)$ that works for all graphs $G$ (or special family of graphs) and that produces numbers satisfying the inequality $\alpha(G^{\boxtimes n}) \leq \psi(G)^{n} \leq \psi(G^{\boxtimes n}) \leq \vartheta(G)^{n}$?