Gaboriau's fixed price conjecture

Let $\Gamma$ be a countable group acting ergodically and essentially freely, by measure-preserving Borel automorphisms, on a standard Borel probability space $X$. Let $\operatorname{Cost}(\Gamma,X)$ be the cost of the induced orbit-equivalence relation. Gaboriau's fixed price conjecture. For fixed $\Gamma$, the number $\operatorname{Cost}(\Gamma,X)$ is the same for all such actions. The notes present this as Gaboriau's conjecture and give no resolution.