When joining to do UPDATEs, always update the table alias not the table name itself.  Otherwise you could have the same problem (or other problems).

UPDATE ...

Specifically what row selection conditions do you want?  Your sample is not clear.  But you almost certainly don't need CASE in any event.

It sounds like you may actually...

Could you provide actual test data, as CREATE TABLE and INSERT statements, rather than just a picture of data.  We can't write code against a picture .🙂

There's no need to join the group row to itself, so this is another possibility:

SELECT c.*
FROM #companies c
LEFT OUTER JOIN #companies g
 ...

If the values are integers, I don't see the need to involve date conversions/char stringing at all:

SELECT yyyymm1, yyyymm2, (yyyymm2 / 100 * 12 +...

For those indexes, you could drop both index1 and index2.

/* this block should only get executed on the first iteration */
DECLARE @greeting VARCHAR(100);
SET @greeting = 'Hello';
/* ==================== */
/* I want this block...

FYI, fwiw, there's some slight differences in the sample data being used and the original sample data picture.  Here's the adjusted data:

INSERT INTO...

I allow my instances to auto-parameterize, so if it's that valuable to the plan, presumably SQL will do so anyway.

Again, with dynamic SQL specifically -- which this code...