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The apparent tensions emerging from the comparison of experimental data of the anomalous magnetic moments of the muon and electron to the Standard Model predictions ($Δa_{μ,e}$) could be interpreted as a potential signal of New Physics. Models encompassing a light vector boson have been known to offer a satisfactory explanation to $Δa_μ$, albeit subject to stringent experimental constraints. Here we explore a minimal extension of the Standard Model via a leptophilic vector boson $Z^\prime$, under the hypothesis of strictly flavour-violating couplings of the latter to leptons. The most constraining observables to this ad-hoc construction emerge from lepton flavour universality violation (in $Z$ and $τ$ decays) and from rare charged lepton flavour violating transitions. Once these are accommodated, one can saturate the tensions in $Δa_μ$, but $Δa_{e}$ is predicted to be Standard Model-like. We infer prospects for several observables, including leptonic $Z$ decays and several charged lepton flavour violating processes. We also discuss potential signatures of the considered $Z^\prime$ at a future muon collider, emphasising the role of the $μ^+μ^- \toτ^+τ^- $ forward-backward asymmetry as a key probe of the model.