Does a monk with Unarmored movement still have to make strength checks to “run”/”climb” a wall?

If a Level 9 (or higher) monk, with the Unarmored Movement feature, wanted to climb a 1000-ft sheer cliff, would they have to make strength checks every turn to see if they continue climbing, fall, or stay where they are? Or would they just keep going without making checks?

(Related question.)