Apparently, the problem was an inconsistent version of NTLM between servers.
After entering the Azure role and disabling the "Require NTLMv2 security" option for clients, it worked:

(Thanks this answer and this answer for inspiration.)
We are currently seeing if we can upgrade our SMTP server for NTLMv2 compatibility. Otherwise, we will have to configure some automated code to somehow disable this parameter for each instance of the role created.
Apparently, this code worked last month. Therefore, I assume that the updated version of Azure OS has changed the default settings.
FYI: registry key for this parameter
[HKEY_LOCAL_MACHINE \ SYSTEM \ CurrentControlSet \ Control \ LSA \ MSV1_0] "NtlmMinClientSec" = DWORD: 20000000
To automate the configuration of a registry key, add a launch task that contains the reg add command as follows:
reg add HKLM\SYSTEM\CurrentControlSet\Control\Lsa\MSV1_0 ^ /v NtlmMinClientSec ^ /t REG_DWORD ^ /d 0x20000000 ^ /f
where /f forcibly overwrites the current setting and ^ just allows you to split the command into several lines for better readability. Also, do not forget to save the command in ASCII encoding to prevent problems when starting the role .
Mike asdf
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