Can Space X Integrate Autorotation in Reusable Rockets?

[Abstractness]

Space X wants to build such a reusable rocket:
Why don't they instead use a foldable rotor and land the thing with autorotation?
I mean it wouldn't take any fuel.

 

[etudiant]

There was one contender that had such a design, but they dropped out early in the competition.
Do note that the rotor idea has some disadvantages, as it adds the dead weight of large hinged blades and their attachments, plus unpowered it has very limited maneuverability.
The Space X reusable booster requires very little fuel, as the empty booster is a fraction of the weight of the full unit. Throttling the engine efficiently is the biggest challenge, but if that is achieved, landing for reuse is a realistic option.