Application Flow Stuck Unless Connection is Explicitly Closed

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* **asyncpg version**: 0.30.0
* **PostgreSQL version**: 16
* **Do you use a PostgreSQL SaaS? If so, which?**: Yes, I am using NeonDB
* **Can you reproduce the issue with a local PostgreSQL install?**: Didn't attempt it.
* **Python version**: 3.11.0
* **Platform**: Windows
* **Do you use pgbouncer?**: No
* **Did you install asyncpg with pip?**: Yes
* **If you built asyncpg locally, which version of Cython did you use?**: [Your Cython version if applicable]
* **Can the issue be reproduced under both asyncio and
[uvloop](https://github.com/magicstack/uvloop)?**: [Yes/No] I did not try uvloop

I'm encountering an issue where the flow of my application gets stuck unless the database connection is explicitly closed after a query, even though it should be managed automatically by the context manager.

**Code Example**:
```python
async def read_record(self, request_details: dict):
"""
Retrieve records based on conditions and request_type.

Args:
request_details (dict): The conditions including request_type.

Returns:
list: The records that match the conditions.
"""
table = self.get_table_name(request_details['request_type'])
conditions = {key: request_details[key] for key in request_details if key != 'request_type'}
condition_str = " AND ".join(f"{key} = ${i+1}" for i, key in enumerate(conditions.keys()))
query = f"SELECT * FROM {table} WHERE {condition_str};"

async with self.get_connection() as conn:
conn.add_log_listener(lambda connection, message: logger.info("PostgreSQL log message: %s", message))
result = await conn.fetch(query, *conditions.values())
logger.info("Fetched records: %s", result)
# await conn.close() # get's stuck without this line

return result
```

Start with the provided asyncpg 0.30.0 example and reproduce it on Python 3.11, PostgreSQL 16, and Windows, first confirming whether the issue occurs with a local PostgreSQL installation. Investigate the behavior of the async context manager around `self.get_connection()` and compare it with the explicit `conn.close()` case; done means the flow completes without requiring that explicit close.

Written by the indexing model from the issue text.