def METHOD_NAME(
        schema,
        table_name,
        column_name,
        engine,
        depth=0,
        type_inference_dag=None,
        metadata=None,
        columns_might_have_defaults=True,
):
    """
    Attempt to cast the column to the best type for it.
    Returns the resulting column type's class.
    Algorithm:
        1. Check for any data in the column.
           - If the column is empty, return the column's current type
             class.
        2. reflect the column's type class.
        3. Use _get_type_classes_mapped_to_dag_nodes to map it to a
           TYPE_INFERENCE_DAG key.
        4. Look up the sequence of types referred to by that key on the
           TYPE_INFERENCE_DAG.
           - If there's no such key on the TYPE_INFERENCE_DAG dict, or if
             its value is an empty list, return the current column's type
             class.
        5. Iterate through that sequence of types trying to alter the
           column's type to them.
           - If the column's type is altered successfully, break
             iteration and return the output of running infer_column_type
             again (trigger tail recursion).
           - If none of the column type alterations succeed, return the
             current column's type class.
    """
    metadata = metadata if metadata else get_empty_metadata()
    if type_inference_dag is None:
        type_inference_dag = TYPE_INFERENCE_DAG
    if depth > MAX_INFERENCE_DAG_DEPTH:
        raise DagCycleError("The type_inference_dag likely has a cycle")
    type_classes_to_dag_nodes = _get_type_classes_mapped_to_dag_nodes(engine)
    column_type_class = _get_column_class(
        engine=engine,
        schema=schema,
        table_name=table_name,
        column_name=column_name,
        metadata=metadata,
    )
    table_oid = get_oid_from_table(table_name, schema, engine)
    column_contains_data = determine_whether_column_contains_data(
        table_oid, column_name, engine, metadata
    )
    # We short-circuit in this case since we can't infer type without data.
    if not column_contains_data:
        return column_type_class
    # a DAG node will be a DatabaseType Enum
    dag_node = type_classes_to_dag_nodes.get(column_type_class)
    logger.debug(f"dag_node: {dag_node}")
    types_to_cast_to = type_inference_dag.get(dag_node, [])
    column_attnum = get_column_attnum_from_name(table_oid, column_name, engine, metadata)
    for db_type in types_to_cast_to:
        try:
            with engine.begin() as conn:
                alter_column_type(
                    table_oid,
                    column_attnum,
                    engine,
                    conn,
                    db_type,
                )
            logger.info(f"Column {column_name} altered to type {db_type.id}")
            column_type_class = METHOD_NAME(
                schema,
                table_name,
                column_name,
                engine,
                depth=depth + 1,
                type_inference_dag=type_inference_dag,
                metadata=metadata
            )
            break
        # It's expected we catch this error when the test to see whether
        # a type is appropriate for a column fails.
        except DatabaseError:
            logger.info(
                f"Cannot alter column {column_name} to type {db_type.id}"
            )
    return column_type_class