  The transport properties of iron under Earth's inner core conditions are
essential input for the geophysical modelling but are poorly constrained
experimentally. Here we show that the thermal and electrical conductivity of
iron at those conditions remains high even if the electron-electron-scattering
(EES) is properly taken into account. This result is obtained by ab initio
simulations taking into account consistently both thermal disorder and
electronic correlations. Thermal disorder suppresses the non-Fermi-liquid
behavior of the body-centered cubic iron phase, hence, reducing the EES; the
total calculated thermal conductivity of this phase is 220 Wm$^{-1}$K$^{-1}$
with the EES reduction not exceeding 20%. The EES and electron-lattice
scattering are intertwined resulting in breaking of the Matthiessen's rule with
increasing EES. In the hexagonal close-packed iron the EES is also not
increased by thermal disorder and remains weak. Our main finding thus holds for
the both likely iron phases in the inner core.
