You must make it to the end of Wave # in order for that objective to be complete. If the scenario ends before that wave, the objective is not considered to be completed. Objectives that state "when the scenario ends" however, may be completed even if the scenario is ended before the final wave.

Note that as long as the game ends in Wave #, no matter the reason (unless a loss condition is triggered), that is considered the end of that wave. 

Example: Brawnen Chapter 1: Ignition

The first two listed objectives both require certain heir spires to be defeated at the end of Wave 4. However, the third objective of defeating the heir fortress gate can cause the scenario to end before Wave 4. If you were to defeat the heir fortress gate in Wave 3, you would not have made it to Wave 4, and therefore would be unable to complete objectives 1 and 2, even if all the relative spires were already defeated.

If you were already in Wave 4 and ended the wave early by defeating the heir fortress gate, then it would count as the "end of Wave 4," meaning objectives 1 and 2 could then be completed if you destroyed the relevant spires.