### Description

Going to Airflow web UI, Browse>DAG Run displays the list of runs, but there is no way to search or filter based on the text in the "Note" column.

### Use case/motivation

It is possible to do a free text search for the "Run Id" field. The Note field may contain pieces of information that may be relevant to find, or to filter on the basis of these notes.

### Related issues

Sorting by Note in List Dag Run fails: 
https://github.com/apache/airflow/issues/30041

### Are you willing to submit a 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)

"""Process form data."""
        conn_id = form.data["conn_id"]
        conn_type = form.data["conn_type"]
        
        
        
        extra = {}
        extra_json = form.data.get("extra")
        if extra_json:
            try:
                extra.update(json.loads(extra_json))
            except (JSONDecodeError, TypeError):
                flash(
                    Markup(
                        "<p>The <em>Extra</em> connection field contained an invalid value for Conn ID: "
                        "<q>{conn_id}</q>.</p>"
                        "<p>If connection parameters need to be added to <em>Extra</em>, "
                        "please make sure they are in the form of a single, valid JSON object.</p><br>"
                        "The following <em>Extra</em> parameters were <b>not</b> added to the connection:<br>"
                        "{extra_json}"
                    ).format(conn_id=conn_id, extra_json=extra_json),
                    category="error",
                )
                del form.extra
        del extra_json
        for key, field_name in self._iter_extra_field_names():
            if key in form.data and key.startswith("extra__"):
                conn_type_from_extra_field = key.split("__")[1]