When password write back is enabled, the new password should sync down to on-premises AD as long as the "passwordPolicies" attribute is not set to "None" or "DisablePasswordSync". You can verify if the user's password policies are set to allow password sync using the Azure AD PowerShell module.
Here is an example PowerShell command to check the password policies for a user:
Get-AzureADUser -ObjectId <UserObjectID> | Select-Object PasswordPolicies
If the password policies are not the issue, you can try forcing a password sync by running the following PowerShell command:
Set-ADSyncAADPasswordSyncConfiguration -SourceConnector "AD Connector" -TargetConnector "Azure AD Connector" -AllowPasswordSync $true
This will force a password sync for all users.
Additionally, you may want to check if the Azure AD Connect synchronization service is running and properly configured.