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Pyth, 18 bytes

L?SIb0hys%L2>Bb1
y

Try it online!

-2 thanks to @Erik the Outgolfer.

The script has two line: the first one defines a function y, the second line calls y with the implicit Q (evaluated stdin) argument.

L?SIb0hys%L2>Bb1
L function y(b)
 ? if...
 SIb the Invariant b == sort(b) holds
 0 return 0
 h otherwise increment...
 y ...the return of a recursive call with:
 B the current argument "bifurcated", an array of:
 b - the original argument
 > 1 - same with the head popped off
 L map...
 % 2 ...take only every 2nd value in each array
 s and concat them back together

1