"Trouble comes uninvited."
Similar occurrences often happen in international politics and diplomacy, where political necessity often decides "the truth."
For instance, now, under economic influence, the London Government had to support the Berlin Government, but their strength alone wasn't enough; they had to rope in the French.
Against this backdrop, the "black hand" could only be "Austria." From the perspective of interest, it just so happened that the Vienna Government also had a motive, which made it even harder to clear their name.
But it didn't matter if they could clear their name or not; on the issue of the Prusso-Russian War, the Anglo-Austrian two countries had already taken opposite stands, and being in too much debt didn't weigh them down.
Franz was indifferent, but Alexander II had a headache. Once the British made their move, most of their diplomatic efforts went down the drain.