Beck's bounded-move conjecture for complete-graph targets

Prove or disprove that for every integer t\ge 3 there exists a constant c_t, depending only on t, such that L(K_n,K_t)\le c_t for all sufficiently large n.

Background

The extremal function L(\mathcal{B},H) is the minimum number of moves in which Player 1 can force a copy of H on the board \mathcal{B}, regardless of Player 2's strategy. The conjecture concerns the target cliques K_t on the single complete board K_n.

The conjecture is known for t=3 and t=4, whereas for t\ge 5 the paper reports only the essentially Turán-theoretic upper bound. The authors note that even the case t=5 has been regarded as particularly difficult.