Does (TaSe₄)₂I really harbor an axionic charge density wave?

A recent experimental work has reported an excess of the non-linear conductivity in the charge density wave (CDW) sliding mode of the quasi one-dimensional compound (TaSe₄)₂I, when a magnetic field is applied co-linearly to the electric field [Gooth et al., Nature 575, 315 (2019)]. This result has opened a conceptual approach, where the CDW gap in (TaSe₄)₂I is opened between Weyl fermions of opposite chirality with the assumption that this compound is a Weyl semi-metal in its undistorted high temperature phase. We report measurements in the sliding state of (TaSe₄)₂I performed in similar conditions. We have found no increase in the magnetoconductivity. In our attempts for understanding this unsettling discrepancy, we stress the specific nature of the Peierls transition in (TaSe₄)₂I and the strong electron-phonon coupling present in this compound. Given the lack of further evidence, we think that it is premature to assert that (TaSe₄)₂I is an axionic insulator.