I believe that the code specified for a specific task will be executed in one thread, but there may not be continuations. It would be very difficult to write valid code if the tasks were opaque to the flow inside the body of the task, so I am sure that you are fine.
On the other hand, your code can become cleaner if you explicitly use the culture where necessary. This may not be feasible depending on the side of ASP.NET, but it is worth considering if it is possible.
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