How to fill NULL values in a PostgreSQL column using the last non-NULL value

I have a PostgreSQL table with the following structure:

CREATE TABLE cte1 (
    entity_id INT,
    assignedtogroup INT,
    time BIGINT
);

INSERT INTO cte1 (entity_id, assignedtogroup, time)
VALUES
    (1, 435198, 1687863949740),
    (1, 435198, 1687863949741),
    (1, NULL, 1687863949742),
    (1, NULL, 1687863949743),
    (1, 435224, 1687863949744),
    (1, 435224, 1687863949745),
    (1, 435143, 1687863949746),
    (1, 435143, 1687863949747),
    (1, 435191, 1687863949748),
    (1, NULL, 1687863949749),
    (2, 435143, 1690452125291),
    (2, 435143, 1690452125292),
    (2, 435191, 1690452125293),
    (2, NULL, 1690452125294);

I would like to fill the empty values in the assignedtogroup column using the previous row's (time just before the current row and same entity_id) non-null value. The expected result should be:

Is there a way to achieve this using only a SELECT statement?

I tried using the LAG function:

SELECT
    entity_id,
    COALESCE(
        assignedtogroup,
        LAG(assignedtogroup) OVER (PARTITION BY entity_id ORDER BY time)
    ) AS filled_assignedtogroup
FROM cte1;

However, I still have a NULL value and for the entity_id 2, the values are completely mixed.

You can find the DB Fiddle : https://www.db-fiddle.com/f/m52Rgq8jtK85g9yvaDMJqz/3

You would be better-off imo using a simple correlation here:

select Entity_Id, Coalesce(assignedtogroup, (
  select assignedtogroup 
	from cte1 cte2 where cte2.entity_id = cte1.entity_id 
	  and cte2.time < cte1.time 
	  and cte2.assignedtogroup is not null 
	  order by time desc
    limit 1
  )), time
from cte1;

Updated DB fiddle https://www.db-fiddle.com/f/m52Rgq8jtK85g9yvaDMJqz/3

you can use a MAX and COUNT window function

Query #1

WITH CTE AS (
    SELECT *,
           COUNT(CASE WHEN assignedtogroup IS NOT NULL THEN 1 END) OVER(
               PARTITION BY entity_id 
               ORDER     BY time
           ) AS rn
    FROM cte1
)
SELECT entity_id, MAX(assignedtogroup) OVER(PARTITION BY entity_id, rn) AS value_ , time
FROM CTE
ORDER BY 1,3;

View on DB Fiddle