A proof of the Bunkbed conjecture on the complete graph for $p\geqslant1/2$

The bunkbed of a graph $G$ is the graph $G\times\left\{ 0,1\right\}$. It has been conjectured that in the independent bond percolation model, the probability for $\left(u,0\right)$ to be connected with $\left(v,0\right)$ is greater than the probability for $\left(u,0\right)$ to be connected with $\left(v,1\right)$, for any vertex $u$, $v$ of $G$. In this article, we prove this conjecture for the complete graph in the case of the independent bond percolation of parameter $p\geqslant1/2$.