I know they moved into the cottage, that was part of what you deem media speculation. They moved when they did so that their children could begin their school year at their new school.
But it was also part of media speculation that it was never intended that they would remain there until William succeeded to the throne. It's too small. They have no space for live-in staff. They can't even have overnight guests. When they have to be away overnight, they'll have to play musical bedrooms, doubling up the children or allowing whoever is watching the children to use theirs. Okay for the short-term, not okay for ten to twenty years.
I've given reasons why I believe the media speculation. If you want to change my mind, give me reasons why not to believe it, why a family of five with their responsibilities would be able to function in that space.
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