The Take-back counting algorithm
05AB1E, 5 bytes

I am not 100% certain this works, but it passes all test cases and I couldn't find any situation where it fails.

Ú¥1QP


Try it online!

Ú¥1QP   Main link. Argument a
Ú       Reverse uniquify a, keeps only last occurence of each element
 ¥      Get all deltas - all 1 if ascending list
  1Q    Compare all deltas to 1
    P   Product of all results