Uniqueness of solution to scalar BSDEs with $Lexp{left(mu sqrt{2log{(1+L)}},right)}$-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 $Lexp{left(mu sqrt{2log{(1+L)}},right)}$-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.