### 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)

return 1
    @task_decorator
    def mytask():
        return 1
    with dag_maker() as dag:
        t1 = setup_task()
        t2 = teardown_task()
        t3 = teardown_task2()
        t4 = teardown_task3()
        with t1 >> t2:
            with t3:
                with t4:
                    mytask()
    assert {t.task_id for t in dag.teardowns} == {"teardown_task", "teardown_task2", "teardown_task3"}
    assert {t.task_id for t in dag.tasks_upstream_of_teardowns} == {"setup_task", "mytask"}
@pytest.mark.parametrize(
    "start_date, expected_infos",
    [
        (
            DEFAULT_DATE,
            [DagRunInfo.interval(DEFAULT_DATE, DEFAULT_DATE + datetime.timedelta(hours=1))],
        ),
        (
            DEFAULT_DATE - datetime.timedelta(hours=3),
            [
                DagRunInfo.interval(
                    DEFAULT_DATE - datetime.timedelta(hours=3),
                    DEFAULT_DATE - datetime.timedelta(hours=2),
                ),