[BUG] Play Games Support ID not sync'ed between native GMS & microg devices

A new Google Play Games Support ID gets generated each time a user decides to clear his account data (within GPlay Games) & creates a new profile (even if associated with the same email address).
While this is working properly for the Google-native version - meaning a new Support ID gets generated, this new ID is not propagated to accounts managed in microg.
In practice:
I have tested this on 5 devices (Android 13-16) until now - new Support IDs get correctly picked up by other Google-native (Play Services, Play Games) devices when syncing the accounts in Android settings.
In contrast, microg gets stuck with the first Support ID it will every pick up associated with an email in particular. This will persist across firmware flashes. Tested on 2 devices (Android 14 & Android 15)
Save data gets properly syncd, but the Support ID does not.
This means that some games which check for Support ID detect the inconsistency between Support IDs as data coming from other accounts.

As a reminder, we are talking about the same Google email across all 5 devices.
The 3 native GMS devices have the same Support ID.
The 2 microg devices have Support IDs different from the native devices, but the same on both microg devices.

n.b. The "sync account" is always greyed out in Android settings for these accounts/devices where I'm using microg.

Devices tested:
Xperia XZ2 / Android 15 / microg 0.3.15.250932 / Play Games Support ID: 56B2E762
Redmagic 6 / Android 14 / microg 0.3.15.250932 / Play Games Support ID: 56B2E762
Xiaomi Mix Fold 2 / Android 13 / stock/native GMS / Play Games Support ID: 3A99DA25
Samsung Z Fold 3 / Android 15 / stock/native GMS / Play Games Support ID: 3A99DA25
Honor Magic V3 / Android 16 / stock/native GMS / Play Games Support ID: 3A99DA25

Start by reproducing the account-data reset and new Play Games profile on the tested Android and microG versions, then compare the Support ID across native GMS and microG devices. Check Android account-sync behavior, noting that sync is reported as greyed out on microG devices. Done means microG receives the new Support ID for the same email after the profile is recreated and no longer retains the original ID across devices or firmware flashes.

Written by the indexing model from the issue text.