The constant-bias Client path conjecture

Let $K_n$ be the complete graph on $n$ vertices, let $P_k$ be the path on $k$ vertices, and write $CW(K_n,P_k,Cn)$ for the Client-Waiter game with bias $Cn$ in which Client aims to claim a copy of $P_k$. Constant-bias path conjecture. For every positive integer $k$ and every constant $C>0$, Client wins

$$CW(K_n,P_k,Cn),$$

provided $n$ is large enough. This would provide a matching lower bound in the fixed-$k$ regime for the upper bound on Client's longest path, complementing the paper's results on the large component and path games.