Let $$\text{EXACT-INDSET} = \{\text{the largest independent set in G has size exactly k}\}.$$ We know that $\text{EXACT-INDSET}$ is in both $\Sigma_2^P$ and $\Pi_2^P$.
My question is what would be the consequences if we prove that EXACT-INDSET is complete in $\Pi_2^P$? Is it would imply that $\Sigma_2^P = \Pi_2^P$, as it is already known that EXACT-INDSET is in both classes?
