### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

The default is ``True``, but subdags will ignore this value and always
        behave as if this is set to ``False`` for backward compatibility.
        Example: A DAG is scheduled to run every midnight (``0 0 * * *``). If
        ``earliest`` is ``2021-06-03 23:00:00``, the first DagRunInfo would be
        ``2021-06-03 23:00:00`` if ``align=False``, and ``2021-06-04 00:00:00``
        if ``align=True``.
        """
        if earliest is None:
            earliest = self._time_restriction.earliest
        if earliest is None:
            raise ValueError("earliest was None and we had no value in time_restriction to fallback on")
        earliest = timezone.coerce_datetime(earliest)
        latest = timezone.coerce_datetime(latest)
        restriction = TimeRestriction(earliest, latest, catchup=True)
        
        
        
        
        
        
        if self.is_subdag:
            align = False
        try:
            info = self.timetable.next_dagrun_info(
                last_automated_data_interval=None,
                restriction=restriction,
            )
        except Exception:
            self.log.exception(
                "Failed to fetch run info after data interval %s for DAG %r",