[release/13.2] Update Azure.Core to latest version - lift all runtime dependencies to latest#14467

Merged

eerhardt merged 5 commits intorelease/13.2from

backport/pr-14361-to-release/13.2

Feb 12, 2026

Commits on Feb 12, 2026

Update to Azure.Core 1.51.1

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Update AzureDeployerTests to use WaitForShutdown instead of StopAsync

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Fix AuxiliaryBackchannelTests by adding a Task that completes when the AuxiliaryBackchannelService is listening and ready for connections.

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Remove double registration of AuxiliaryBackchannelService as an IHostedService.

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Fix ResourceLoggerForwarderServiceTests to ensure the ResourceLoggerForwarderService has started before signalling the stopping token.

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