Probabilistic threshold conjecture for $k$-SAT games

Let $\mathcal{C}_n^{(k)}$ be the $k$-uniform clause set on $n$ Boolean variables, and let $\mathcal{F}_{SAT}$ be the family of satisfiable clause sets. Consider the $(1:q)$ Waiter–Client and Client–Waiter $k$-SAT games, and denote their threshold biases by $b_{\mathcal{F}_{SAT}}^{WC}$ and $b_{\mathcal{F}_{SAT}}^{CW}$, respectively.

$k$-SAT threshold conjecture. The threshold biases satisfy

$$\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{F}_{SAT}}^{WC})^{-1}}{\ln 2}\right\}=\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{F}_{SAT}}^{CW})^{-1}}{\ln 2}\right\}=1.$$

The conjecture predicts the probabilistically suggested asymptotic threshold for both game orientations. The paper proves the threshold only up to exponential and polynomial factors in $k$ for the Waiter–Client and Client–Waiter versions, respectively, so the exact asymptotic remains open.