The giant component size conjecture in the Waiter–Client game

Let $K_n$ be the complete graph on $n$ vertices, and let $\mathcal L(n,q)$ denote the largest component size that Waiter can force in Client's graph when playing a $(q:1)$ Waiter–Client game on $E(K_n)$. Giant component size conjecture. For any constant $\varepsilon>0$ and sufficiently large $n$, if $q=(1-\varepsilon)n$, then

$$\mathcal L(n,q)=\min\{n,2(n-q-1)\}.$$

This conjecture concerns the sharp value of the largest component in the supercritical regime. The paper proves a lower bound of $2\varepsilon n-2$ for the largest component when $q\leq(1-\varepsilon)n$, and the authors believe that the bound in the corresponding theorem is sharp.