RU-ALLRUSSIAN 2018 Problem 94

On the $n\times n$ checker board, several cells were marked in such a way that lower left ( $L$ ) and upper right( $R$ ) cells are not marked and that for any knight-tour from $L$ to $R$ , there is at least one marked cell. For which $n>3$ , is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?