I think I can not answer your question, but I have three ideas that might be useful:
1. Cluster randomized trials often ask similar questions. Having a look at CRT methods may suggest you a solution.
2. Similar questions may arise in time series analysis. I wouldn't be surprised if a standard tool for time series could solve your problem.
3. I like maximum likelihood methods so I find it interesting to treat the question as minimising three likelihood functions. You could use the likelihood-ratio test to test significance.
I hope you find something useful from the above.
Gábor
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