He’ll simply refuse to let you continue, won’t give you a spirit orb or anything else. You have to go back and download the rune, otherwise the shrine won’t complete.
Whether or not you actually USED the rune to solve the shrine puzzles, is something the monk doesn’t care for. But you have to download it anyway.
How do you even pull that off lol
The easiest way is with the Magnesis Shrine - all of the cracked walls/cubes can be destroyed with an Iron Sledgehammer, and there is one inside the Stasis Shrine and another one on a ledge outside of it. You have to toss the hammer at the cubes in the last part, but you can just pick it up again after climbing the ladder. No glitches / exploits needed ;)
In the other shrines, you need glitches, exploits or other runes. In the Stasis Shrine for example, you can use magnesis and the Iron Sledgehammer to push both the stone orb rolling towards you and the one blocking the tiny bridge away.