Here's the solution, in case anyone needs it:
Cerff gur gbc yrsg enoovg, cerff gur obggbz evtug enoovg, naq cerff gur enoovg gung vf gjb ubyrf yrsg bs gur obggbz evtug enoovg.
(You can use a rot13 decoder to read the solution.)
Viewing post in Tale of the Bear and the Beech Tree comments
If you want to figure out the solution for yourself, here's the reasoning process:
When you click on any one of the seven rabbit holes, a certain set of rabbits flips their state. Write out those seven possible patterns. Then consider the top-left rabbit: There are only two ways to get this rabbit to emerge. Likewise, consider the bottom-right rabbit: there are only two ways to get this rabbit to emerge, as well.
If you stare at the seven patterns for a second, you should be able to see the solution. There are only four possibilities to consider, and one of them works.