A good jiggle on the rope from the bottom could have dislodged it. You just have to give it the right force and make it pop off from the rail.

Don't get me started with the grappling hook. After that scene happened, the remainder of the movie I expected this to come back and bite him in the ass. Nope. Nothing. What's worse was that there was really no way of getting rid of the grappling hook. He couldn't have jumped up to get it because it was too high which is why he needed the grappling hook in the first place. And then he needed it to get down, so he couldn't have gotten rid of it there either.

A grapple that supported a man's body weight just magically disappeared from the top of a building and ended up in a nice bundle by the front gates? I guess we can't put anything past Louis Malle; after all, he did make Zazie dans le Metro a couple years later!
~ Observe, and act with clarity. ~

Symbolism? I don't associate his getting into the elevator with his execution. AS for the rope falling to the street, I just thought that it was amusing that this would have happened, especially since Tavernier's troubles began when he went back to get that rope. After all the effort he put into getting that rope back, it just fell down on its own. Excellent dark comedy!
~~
JimHutton (1934-79) and ElleryQueen

Spoilers.
There are two things that are not fully explained but I think you have to draw your own conclusions with what you are given. In many other reviews, including Wikipedia, it is mentions "he dealt with the rope," and then got stuck in the elevator. But if you carefully watch the elevator scenes - you can see when the three go down the elevator - the button to the ground floor is pushed - they are on the 10th floor the first floor is really the second. And they go down. When Julien goes back in - it is clear that he selects the 10th floor and he is on his way up when the elevator stops around the 9th floor in between floors. He never makes it back up to retrieve the rope and when he finally gets out, he doesn't check for the rope. What he thought would get him caught actually didn't. I think you have to figure the rope got disentangled from the railing and fail to the street. The child retrieved it and it resolved itself without any repercussions, which is brilliant since getting back in the elevator and going up symbolized the lift to the scaffold, his execution. And that brings the second thing - is the ending - Lino Ventura (the detective) tells you what punishment everyone would get doing either crime (the husband or the German couple). That part really doesn't make sense. Julien would be executed for the German couple murder because he doesn't have an alibi, but his confession to murdering her husband would get him a reduced sentence of 5 years where she might get much more. And it comes full circle to the beginning of their love and wanting to be together. But it is Jeanne Moreau's soliloquy at the very end that is so beautiful and these questions resolve itself because it allows her this soliloquy. This is a fantastic movie.

It fell down.
You're right to be confused, however, as if you refer to the shot (at 14:27) when Julien looks back up at the building to see that he's forgotten the rope, you'll notice that it's actually hanging on the middle of three tiered balconies. So if the rope were to fall off the railing that night, if would only have landed on the balcony ground a few feet below it.

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