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The possibility of explaining shear flow dynamos by a magnetic shear-current (MSC) effect is examined via numerical simulations. Our primary diagnostics is the determination of the turbulent magnetic diffusivity tensor $\boldsymbolη$. In our setup, a negative sign of its component $η_{yx}$ is necessary for coherent dynamo action by the SC effect. To be able to measure turbulent transport coefficients from systems with magnetic background turbulence, we present an extension of the test-field method (TFM), applicable to our setup where the pressure gradient is dropped from the momentum equation: the nonlinear TFM (NLTFM). Our momentum equation is related to Burgers' equation and the resulting flows are referred to as magnetized burgulence. We use both stochastic kinetic and magnetic forcings to mimic cases without and with simultaneous small-scale dynamo action (SSD). When we force only kinetically, negative $η_{yx}$ are obtained with exponential growth in both the radial and azimuthal mean magnetic field components. Using isotropic magnetic forcing, the field growth is no longer exponential, while NLTFM yields positive $η_{yx}$. By employing an alternative forcing from which wavevectors having small components are removed, the exponential growth is recovered, but the NLTFM results do not change significantly. Analyzing the dynamo excitation conditions for the coherent SC and incoherent $α$ and SC effects shows that the incoherent effects are the main drivers of the dynamo in the majority of cases. We find no evidence for MSC-effect-driven dynamos in our simulations.