<think>
Okay, let's see. The user wants me to fix the limit_map function. The problem is that the current implementation of limit_map is only keeping the keys where the value is less than or equal to the ceiling. But according to the description, limit_map should bring every value in the store down to the ceiling, but drop the names whose values were over the ceiling. Wait, the original problem says that held_down returns the value if it's already under the ceiling, otherwise returns the ceiling. So, for limit_map, it should apply held_down to each value, but then remove the keys where the original value was over the ceiling. Wait, no. Let me recheck.

The problem says: "limit_map brings every value of a store down that way, and it drops the names whose values were over the ceiling instead of bringing them down." So, the process is: for each value in the store, apply held_down (so if the value is above the ceiling, it's set to ceiling; otherwise, it's the same). But then, the names (keys) whose values were over the ceiling are dropped. Wait, that's a bit ambiguous. Wait, the original description says that held_down returns a value brought down to a ceiling, leaving a value already under it alone. So, if the value is already under the ceiling, it's left as is. But for limit_map, the task is to bring every value down, but then drop the names (keys) whose values were over the ceiling. Wait, maybe the names (keys) that had values over the ceiling are removed from the output. So, for each key in the store, we check if the original value was over the ceiling. If it was, then we don't include that key in the output. Otherwise, we include it with the held_down value.

Wait, the current code for limit_map is:

def limit_map(store: dict, ceiling: int) -> dict:
    out = {}
    for key, value in store.items():
        if value <= ceiling:
            out[key] = value
    return out

But according to the problem, the correct behavior is that for each value, we apply held_down (so if the value is above the ceiling, it's set to ceiling). But then, the keys whose original values were over the ceiling are dropped. Wait, that's not what the current code does. Because the current code only adds the key to the output if the original value is <= ceiling. But if the original value is above the ceiling, then the held_down would set it to ceiling, but the current code is not adding that key. So the current code is not doing the correct thing.

Wait, the problem says that limit_map should bring every value down to the ceiling, but drop the names (keys) whose values were over the ceiling. So, the process is: for each key in the store, apply held_down to the value. Then, if the original value was over the ceiling, we drop the key. Otherwise, we keep it. Wait, but that's not correct. Because if the original value was over the ceiling, then held_down would set it to the ceiling. But the problem says that the names (keys) whose values were over the ceiling are dropped. So, even if the value is brought down, if the original value was over the ceiling, the key is dropped.

Wait, the problem says: "it drops the names whose values were over the ceiling instead of bringing them down." So, the names (keys) whose values were over the ceiling are not included in the output. So, even if the value is brought down, if the original value was over the ceiling, the key is excluded.

So, the correct approach is