Separation exponent at least $2.085$ via eight tournament layers

For every real $\varepsilon>0$ and every integer $S\ge2$, there exist a natural number $n$ and a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ such that

Here $s(f)$ is the maximum sensitivity over all inputs, and $bs(f)$ is the maximum size of a pairwise disjoint family of nonempty sensitive coordinate blocks, also maximized over inputs. The function and its dimension may depend on $\varepsilon$ and $S$. This establishes a lower bound of $2.085$ on the campaign separation exponent, with arbitrarily large sensitivity.