Italy joins Prussia in the Franco-Prussian War

In our timeline most Italians supported Prussia against Napoleon III and only supported France once the Third Republic was procclaimed.
What if Italy had entered a formal alliance with Prussia, after the Austro-Prussian War, joined Prussia in the Franco-Prussian War and gotten Corsica and Nice?
 
The dilemma of the House of Savoy at this point is whether to ask for Nice and Savoy back along with Corsica, or whether to march into (French-protected) Rome and claim it as their capital. I don't think the other powers would let them choose both Rome and the French-occupied territories.

IOTL, they chose Rome, so it wasn't necessary for them to join the Prussians in their war at this point.
 
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