Markov kk-Menger

The second player has a Markov winning strategy in the game Gfin(KX,KX)\mathsf{G}_{\mathrm{fin}}(\mathcal K_X,\mathcal K_X) (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of DOI 10.1016/j.topol.2021.107772 for more details.