In [3], the authors proved that uniqueness holds among solutions whose exponentials are \(L^p\) with \(p\) bigger than a constant \(\gamma\) (\(p\textgreater{}\gamma\)). In this paper, we consider the critical case: \(p=\gamma\). We prove that the uniqueness holds among solutions whose exponentials are \(L^\gamma\) under the additional assumption that the generator is strongly convex.