Another way to see this is to track how the two cages are meant to specialize at different operating points, by design.
The outer cage (high resistance, low reactance) is built to dominate at STARTING: since reactance is largest at starting and the outer cage's reactance is small to begin with, it ends up with a lower total impedance than the inner cage at that instant, so it carries the larger share of rotor current when the motor is switched on.
The inner cage (low resistance, high reactance) is built to dominate at RUNNING speed: once the motor is up to speed, slip is small, rotor frequency is small, and the high reactance of the inner cage becomes negligible, leaving its low resistance to dominate and carry most of the current efficiently.
This deliberate role-swap is the entire point of the double cage design: a high effective starting resistance (for good starting torque, via the outer cage) combined with a low running resistance (for good efficiency, via the inner cage), something a single cage cannot achieve at both extremes at once.
So at the moment of starting, it is the outer cage that carries the larger portion of the rotor current.
\[ \boxed{\text{At starting, more rotor current flows in the outer cage}} \]