Adopt shared Connect-EXOAdvanced in PublicFolders/MailPublicFolderSync/* scripts

Parent: #2590
Depends on: #2591

Refactor the five root scripts in PublicFolders/MailPublicFolderSync/ to use the shared Connect-EXOAdvanced / Disconnect-EXOAdvanced helper. All five carry the same silent-failure pattern documented in the parent issue.

Scripts

• PublicFolders/MailPublicFolderSync/Sync-ModernMailPublicFolders.ps1 — remove the connect block PR #2589 just added and call the helper instead. Preserves sovereign-cloud support.
• PublicFolders/MailPublicFolderSync/Sync-MailPublicFolders.ps1
• PublicFolders/MailPublicFolderSync/Sync-MailPublicFoldersCloudToOnprem.ps1
• PublicFolders/MailPublicFolderSync/Import-MailPublicFolders.ps1 — also fix unguarded Disconnect-ExchangeOnline at line 244 and line 257 — replace with Disconnect-EXOAdvanced -Connection $exo so we only tear down what we opened.
• PublicFolders/MailPublicFolderSync/Import-PublicFolderMailboxes.ps1

Problem code (representative)

Sync-MailPublicFolders.ps1#L169-L193:

try {
    Import-Module ExchangeOnlineManagement -ErrorAction SilentlyContinue
    if (Get-Module ExchangeOnlineManagement) {
        $connectParams = @{
            ConnectionUri = $ConnectionUri
            Prefix        = "Remote"
            ErrorAction   = "SilentlyContinue"      # 🐛 silent auth failure
        }
        if ($null -ne $Credential) { $connectParams.Credential = $Credential }
        Connect-ExchangeOnline @connectParams
        $script:isConnectedToExchangeOnline = $true # 🐛 set even if Connect failed
    } else {
        Write-Warning $LocalizedStrings.EXOV2ModuleNotInstalled
        exit
    }
} finally { ... }

Target pseudocode

. $PSScriptRoot\..\..\Shared\M365\EXOConnection.ps1

$exo = Connect-EXOAdvanced -ConnectionUri $ConnectionUri `
                           -AzureADAuthorizationEndpointUri $AzureADAuthorizationEndpointUri `
                           -Credential $Credential `
                           -Prefix 'Remote'
if (-not $exo) { return }   # helper printed the reason

try {
    # ...existing sync logic...
} finally {
    Disconnect-EXOAdvanced -Connection $exo   # only disconnects the session we opened
}

Removes:

• The InitializeExchangeOnlineRemoteSession function
• The $script:isConnectedToExchangeOnline state variable
• The Disconnect-ExchangeOnline cleanup block
• The EXOV2ModuleNotInstalled localized string (owned by the helper now)

Validation

• Existing Pester tests for Sync-ModernMailPublicFolders.ps1 must be updated to mock Connect-EXOAdvanced / Disconnect-EXOAdvanced instead of Connect-ExchangeOnline / Import-Module.
• New Pester tests for the other four scripts covering: connect success, connect failure returns early, disconnect only called when connect succeeded.
• Lab test: minimum one full sync run in a real tenant and one in a sovereign-cloud tenant.

Start with Shared/M365/EXOConnection.ps1 and the five scripts under PublicFolders/MailPublicFolderSync/, comparing their current connection and cleanup blocks with the target helper usage. Update the existing Sync-ModernMailPublicFolders.ps1 Pester tests and add coverage for the other four scripts for connection success, early return on failure, and conditional disconnect; done includes the requested full-sync lab checks in standard and sovereign-cloud tenants.