Maximilian I and His Effects on French Relations

I came across a post whining about Timeline-191 and it raised up a certain point. Wouldn't Napoleon III placing an Austrian Archduke on the Throne of Mexico help relations? If Maximilian continued to reign, was deported, or willingly left Mexico how would it relate to Franco-Austrian relations? Would France give support to them against the Prussians? Or will it be like in OTL where they all think Austria will win and the Prussians backstab everybody? More importantly, how did the percieved abandonment of Maximilian by Napoleon in OTL get viewed by the rest of the world?
 
If Max continues to reign Nap will continue to waste French money and lives in Mexico until he gets overthrown with no need of going to war against Prussia, let alone in defense of Austria.

If he gets out alive, I suppose he goes back to Austria and everybody will be too embarrassed to talk about the thing again.
 
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