In what state – solid, liquid, gaseous – does the substance have the highest thermal conductivity?

The substance has the highest thermal conductivity in the solid state.
This is explained by the fact that in solids the atoms are located closer to each other, in a certain order and are strongly connected to each other, make an oscillatory motion. When heat is transferred, the atom begins to increase the amplitude of its vibrations. Since the atoms are linked by a crystal lattice, an increase in the vibration amplitude of one atom leads to an increase in the vibration amplitude of neighboring atoms. The process of transmitting vibrational motion occurs quickly.
In gases and liquids, atoms are at a greater distance and are less connected to each other. The transfer of energy between atoms is slower.