A long, uniform metallic rod of cross-sectional area \(A\) and thermal conductivity \(k\) has its left end (\(x = 0\)) maintained at a constant high temperature \(T_H\) and its right end (\(x = L\)) maintained at a lower temperature \(T_C\). Unlike an ideal insulated rod, the lateral surfaces of this rod are uninsulated, allowing thermal energy to dissipate into the surrounding environment. After the system reaches a steady state, which of the following claims correctly describes how the magnitude of the temperature gradient \(\left|\dfrac{\Delta T}{\Delta x}\right|\) changes along the rod from \(x = 0\) to \(x = L\), and provides a valid justification?