Beck's bounded-move conjecture for strong Ramsey games

Let $K_n$ be the complete graph on $n$ vertices, and let $K_t$ be the complete graph on $t$ vertices. The function $L(K_n,K_t)$ denotes the minimum number of moves required for player $P_1$ to establish a copy of $K_t$ in the strong Ramsey game on $K_n$, regardless of $P_2$'s play.

Beck's conjecture. For sufficiently large $n$ and every integer $t\geq 3$, there is a constant $c_t$, depending only on $t$, such that

$$L(K_n,K_t)\leq c_t.$$

This conjecture asks whether player $P_1$ can always win in a number of moves bounded independently of the board size. Beck identified it as a major open problem, even for $t=5$.