$\mu \to e \gamma$ selecting scalar leptoquark solutions for the $(g-2)_{e,\mu}$ puzzles.

We demonstrate that the chirality enhancement required to simultaneously explain the $(g-2)_{e}$ and $(g-2)_{\mu}$ data with a single scalar leptoquark is in direct conflict with the existing constraint on the induced lepton flavor violation as given by the limits on the branching ratio for the $\mu \to e \gamma$ process. We furthermore investigate all potentially viable combinations of two scalar leptoquarks that can simultaneously address the $(g-2)_{e,\mu}$ discrepancies only to find that the current experimental bound on $\mu \to e \gamma$ represents an unsurmountable obstacle to that appealing prospect, apart from the trivial scenario in which $S_1$ leptoquark would generate chirality enhanced contributions solely to $(g-2)_{e}$ while $R_2$ leptoquark would only affect $(g-2)_{\mu}$ or the other way around. Our results with regard to the simultaneous explanation of observed discrepancies are very robust as they either do not depend on the scale of new physics or, if they do, exhibit only a mild dependence.