A proof of the bunkbed conjecture for the complete graph at $p=\frac{1}{2}$

The bunkbed of a graph $G$ is the graph $G\times K_2$. It has been conjectured that in the independent bond percolation model, the probability for $(u,0)$ to be connected with $(v,0)$ is greater than the probability for $(u,0)$ to be connected with $(v,1)$, 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 $p=1\mathrm{/}2$.