Transport properties in the warm dense matter (WDM) regime, e.g. electrical conductivity, are crucial for accurate modelling of planetary physics and inertial confinement fusion. Recently, it has been suggested that the electrical heating experiments, on which most state-of-the art electrical conductivity tables are built, may contain a systematic error due to the electrothermal instability (ETI). ETI develops in all current heated metals and leads to a thermal non-uniformity which may bias conductivity measurements. We propose to use X-ray absorption spectroscopy at ESRF to obtain the world’s first measurement of this thermal non-uniformity and calculate the existing discrepancy between the current conductivity tables and their correct values. This will significantly improve simulations of planetary interiors and fusion capsules.