SQL Server 2017

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DECLARE @EventsTBL TABLE (PID INT, EID INT, StartDate DATETIME, EndDate DATETIME);
INSERT INTO @EventsTBL VALUES
(13579, 1,  '01 Jan 2018', '31 Mar 2019'),
(13579, 2,  '01 Feb 2018', '31 May 2018'),
(13579, 2,  '01 Jul 2018', '31 Jan 2019'),
(13579, 7,  '01 Mar 2018', '31 Mar 2019'),
(13579, 5,  '01 Feb 2018', '30 Apr 2018'),
(13579, 5,  '01 Oct 2018', '31 Mar 2019'),
(13579, 8,  '01 Jan 2018', '30 Apr 2018'),
(13579, 8,  '01 Jun 2018', '31 Dec 2018'),
(13579, 13, '01 Jan 2018', '31 Mar 2019'),
(13579, 6,  '01 Apr 2018', '31 May 2018'),
(13579, 6,  '01 Sep 2018', '30 Nov 2018'),
(13579, 4,  '01 Feb 2018', '31 Jan 2019'),
(13579, 19, '01 Mar 2018', '31 Jul 2018'),
(13579, 19, '01 Oct 2018', '28 Feb 2019'),
(13570, 16, '01 Feb 2018', '30 Jun 2018'),
(13570, 16, '01 Aug 2018', '31 Aug 2018'),
(13570, 16, '01 Oct 2018', '28 Feb 2019'),
(13570, 23, '01 Mar 2018', '30 Jun 2018'),
(13570, 23, '01 Nov 2018', '31 Jan 2019');

WITH cte1 AS (
    SELECT e.PID, a.EIDCount, StartDate, EndDate
    FROM @EventsTBL AS e
    JOIN (
        SELECT PID, COUNT(DISTINCT EID) AS EIDCount
        FROM @EventsTBL
        GROUP BY PID
    ) AS a ON e.PID = a.PID
), cte2 AS (
    SELECT PID, EIDCount, StartDate AS pdate, 1 AS pval
    FROM cte1
    UNION ALL
    SELECT PID, EIDCount, EndDate, 0
    FROM cte1
    UNION ALL
    SELECT PID, EIDCount , DATEADD(DAY, 1, EndDate), -1
    FROM cte1
), cte3 AS (
    SELECT PID, EIDCount, pdate, SUM(pval) OVER (PARTITION BY PID ORDER BY pdate, pval) AS evtcount
    FROM cte2
), cte4 AS (
    SELECT PID, EIDCount, pdate, MAX(evtcount) AS evtcount
    FROM cte3
    GROUP BY PID, EIDCount, pdate
), cte5 AS (
    SELECT PID, EIDCount, pdate, evtcount, CASE
        WHEN evtcount < EIDCount AND LAG(evtcount) OVER (PARTITION BY PID ORDER BY pdate) < EIDCount THEN 0
        WHEN evtcount = EIDCount AND LAG(evtcount) OVER (PARTITION BY PID ORDER BY pdate) = EIDCount THEN 0
        ELSE 1
    END AS chg
    FROM cte4
), cte6 AS (
    SELECT PID, EIDCount, pdate, evtcount, SUM(chg) OVER (PARTITION BY PID ORDER BY pdate) AS grp
    FROM cte5
)
SELECT PID, MIN(pdate) AS StartDate, MAX(pdate) AS EndDate
FROM cte6
WHERE evtcount = EIDCount
GROUP BY PID, grp
ORDER BY PID, StartDate

PID	StartDate	EndDate
13570	2018-03-01 00:00:00.000	2018-06-30 00:00:00.000
13570	2018-11-01 00:00:00.000	2019-01-31 00:00:00.000
13579	2018-04-01 00:00:00.000	2018-04-30 00:00:00.000
13579	2018-10-01 00:00:00.000	2018-11-30 00:00:00.000