Uniqueness of solution to scalar BSDEs with $L\exp{\left (\mu \sqrt {2\log {(1+L)}}\,\right )} $-integrable terminal values

In [4], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is $L\exp{\left(\mu \sqrt{2\log{(1+L)}}\,\right)}$-integrable with the positive parameter $\mu$ being bigger than a critical value $\mu\_0$. In this note, we give the uniqueness result for the preceding BSDE.