Geometrisation of Ohm's reciprocity relation in a holographic plasma

It has been recently pointed out that the familiar reciprocity relation between the conductivity $\sigma$ and resistivity $\rho$, which I refer to as $\textit{Ohm's reciprocity relation}$, should not be expected to hold in all possible settings, but is rather a property that may (or may not) emerge as a consequence of specific features, or in certain limits of interest, of a given theory. In this work I prove an analogous statement: $\rho= \sigma^{-1}$, across two different classes of holographic theories related by a generalisation of the electric-magnetic duality in the $D=4+1$ bulk. In terms of the dual hydrodynamic theories, this statement is shown to imply the suppression of any contributions to the transport coefficients from dynamical electromagnetic fields, present in only one of the two theories. This makes the two theories, as far as late-time linear electric transport is concerned, equivalent. I then confirm these findings by considering one specific model and run numerical simulations in different settings.