On absolute and unconditional convergence of series in the general Franklin system

We prove that, for any admissible sequence, the corresponding general Franklin system $\{f_n(x)\}_{n=0}^{\infty}$ possesses the following property. A series $\sum_{n=0}^{\infty}a_nf_n(x)$ is absolutely convergent almost everywhere on a set $E$ if and only if it is unconditionally convergent almost everywhere on $E$.