t-SQL to update table to remove overlapping time frames

I was wondering if someone could help me with this SQL statement?

Say, I have an SQL Server 2008 table like this:

id -- INT PRIMARY KEY
dtIn -- DATETIME2
dtOut -- DATETIME2
type -- INT

id  dtIn    dtOut    type
1   05:00   10:00    1
2   08:00   16:00    2
3   02:00   08:00    1
4   07:30   11:00    1
5   07:00   12:00    2

I need to remove any time overlaps in the table above. This can be illustrated with this diagram:

So I came up with this SQL:

UPDATE [table] AS t 
SET dtOut = (SELECT MIN(dtIn) FROM [table] WHERE type = t.type AND t.dtIn >= dtIn AND t.dtIn < dtOut) 
WHERE type = t.type AND t.dtIn >= dtIn AND t.dtIn < dtOut

But it doesn't work. Any idea what am I doing wrong here?

****EDIT****

OK, it took me awhile to get to this. Seems to be a working SQL for what I need it for:

--BEGIN TRANSACTION;

--delete identical dtIn
DELETE dT1
FROM tbl dT1
WHERE EXISTS
(
    SELECT *
    FROM tbl dT2
    WHERE dT1.Type = dT2.Type
    AND dT1.dtIn = dT2.dtIn
    AND (
            dT1.dtOut < dT2.dtOut
            OR (dT1.dtOut = dT2.dtOut AND dT1.id < dT2.id)
        )
);

--adjust dtOuts to the max dates for overlapping section
UPDATE tbl
SET dtOut = COALESCE((
    SELECT MAX(dtOut)
    FROM tbl as t1
    WHERE t1.type = tbl.type
    AND t1.dtIn < tbl.dtOut 
AND t1.dtOut > tbl.dtIn
    ), dtOut);

-- Do the actual updates of dtOut
UPDATE tbl
SET dtOut = COALESCE((
    SELECT MIN(dtIn)
    FROM tbl as t2
    WHERE t2.type = tbl.type AND
          t2.id <> tbl.id AND
          t2.dtIn >= tbl.dtIn AND t2.dtIn < tbl.dtOut
    ), dtOut);

--COMMIT TRANSACTION;