Creating Groups based on if Date Period Connects with Previous Row

I have test data like the following:

DROP TABLE IF EXISTS #test;
CREATE TABLE #test (id INT, start_date DATE,  end_date DATE)
INSERT INTO
    #test
VALUES 
    (1, '2023-01-01', '2024-01-01'),
    (1, '2023-05-01', '2024-07-01'),
    (1, '2025-01-01', '2026-01-01');

SELECT * 
FROM #test;

I am trying to create integer groups that check if the prior row connects with the current. If it doesn't connect, I want to increment the integer to notate that its a separate group.
I have tried below to check if the current row connects with the prior based on if the current period start_date comes before the prior periods end date. this would signal that the time periods connect and be treated as the same group.

WITH connects AS (
    SELECT
        id,
        CASE
            WHEN LAG(id, 1) OVER (PARTITION BY id ORDER BY start_date) = id AND (
                start_date <= LAG(end_date, 1) OVER (PARTITION BY id ORDER BY start_date)
                OR
                end_date <= LAG(end_date, 1) OVER (PARTITION BY id ORDER BY start_date)
            ) THEN 1
            WHEN LAG(id, 1) OVER (PARTITION BY id ORDER BY start_date) IS NULL THEN 1
            ELSE 0
        END AS connects_flag
    FROM
        #test
)
SELECT
    *
FROM
    connects;

Desired outcome:

Here's one way to do it to with cumulative aggregation:
In the common table expression I've used a combination of lag and iif to create a column that will contain 1 whenever the previous row's end date is either missing or smaller than the current row's start date, and 0 otherwise (so 0 when the previous row overlaps with the current, 1 if it doesn't).
Then all that's left to do is to commulatively sum the values of that column.
I've also added some more rows to your sample table:

DROP TABLE IF EXISTS #test;
CREATE TABLE #test (id INT, start_date DATE,  end_date DATE)
INSERT INTO
    #test
VALUES 
    (1, '2023-01-01', '2024-01-01'),
    (1, '2023-05-01', '2024-07-01'),
    (1, '2024-05-01', '2024-07-01'),
    (1, '2024-08-01', '2024-09-01'),
    (1, '2025-01-01', '2026-01-01'),
    (1, '2026-05-01', '2026-07-01');
    

WITH CTE AS
(
    SELECT id
         , start_date
         , end_date
         , IIF(
             ISNULL(
                 LAG(end_date) OVER(PARTITION BY id ORDER BY start_date)
             , '0001-01-01') < start_date
             , 1, 0) As grp
    FROM #test
)

SELECT id
     , start_date
     , end_date
     , SUM(grp) OVER(PARTITION BY id ORDER BY start_date) DateGroup
FROM CTE;

Results:

You can see a live demo on db<>fiddle