AWS: Why does Auto Minor Version Upgrade Flag not upgrade to latest minor Version of Database.

Shout out to Asmita V. for this answer.

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PROBLEM DESCRIPTION
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Why were the RDS PostgreSQL instances not automatically upgraded to the latest Minor Version available even though the Auto Minor Version Upgrade Flag is enabled for your instances.

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RESPONSE
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The reason why your instances was not upgraded to these minor version is that AMVU will only upgrade the engine version for your RDS instance if the current engine version is being deprecated or the new one contains very important cumulative bug fixes and an upgrade is absolutely necessary. 

Please note that while we highly recommend that you perform an upgrade to 10.9, this upgrade will not happen automatically as of now using AMVU as the automatic upgrades happen only when absolutely necessary and you can also view such actions using describe-pending-maintenance-actions command.

If there is an auto minor version upgrade scheduled as a maintenance, please be assured that you will get a separate notification explicitly mentioning the same. Currently, in this case the upgrade will have to be applied manually.

Further, at your end, you can check if the minor version upgrade will happen automatically or not by using the below CLI command:

$aws rds describe-db-engine-versions --output=table --engine postgres --engine-version 10.6

Output:
||+-------------------------------------------------------------------------------------------+||
|||                                    ValidUpgradeTarget                                     |||
||+-------------+---------------------+-----------+----------------+--------------------------+||
||| AutoUpgrade |     Description     |  Engine   | EngineVersion  |  IsMajorVersionUpgrade   |||
||+-------------+---------------------+-----------+----------------+--------------------------+||
|||  False      |  PostgreSQL 10.7-R1 |  postgres |  10.7          |  False                   |||
|||  False      |  PostgreSQL 10.9-R1 |  postgres |  10.9          |  False                   |||
|||  False      |  PostgreSQL 11.1-R1 |  postgres |  11.1          |  True                    |||
|||  False      |  PostgreSQL 11.2-R1 |  postgres |  11.2          |  True                    |||
|||  False      |  PostgreSQL 11.4-R1 |  postgres |  11.4          |  True                    |||
||+-------------+---------------------+-----------+----------------+--------------------------+||

As you can see from the above output, for 10.6 version "AutoUpgrade" column is marked as "False" for all minor version upgrade (either 10.7 or 10.9) . So, it has to be done manually. Please make sure to upgrade to the latest minor version  (10.9)  so that you wont be prone to any security vulnerabilities as per the following notice:

[+] https://www.postgresql.org/about/news/1949/