  In \cite{HuTang2018ECP}, the existence of the solution is proved for a scalar
linearly growing backward stochastic differential equation (BSDE) when the
terminal value is $L\exp\left(\mu\sqrt{2\log(1+L)}\right)$-integrable for a
positive parameter $\mu>\mu_0$ with a critical value $\mu_0$, and a
counterexample is provided to show that the preceding integrability for
$\mu<\mu_0$ is not sufficient to guarantee the existence of the solution.
Afterwards, the uniqueness result (with $\mu>\mu_0$) is also given in
\cite{BuckdahnHuTang2018ECP} for the preceding BSDE under the uniformly
Lipschitz condition of the generator. In this note, we prove that these two
results still hold for the critical case: $\mu=\mu_0$.
