In that situation, if the central bank gives disparate signals to the outside world, the general public does not know at all what it is actually supposed to be supporting. Admittedly, it is comparatively easy to create unanimity in the event of a conflict.
But the central bank’s standing with the general public, which is intended to be a protection in the event of a conflict — that is something which it has to have earned and built up beforehand. And it has that standing only if the general public is aware of what the central bank stands for.