SHARYGIN 2022 Problem 12

Let $K$ , $L$ , $M$ , $N$ be the midpoints of sides $BC$ , $CD$ , $DA$ , $AB$ respectively of a convex quadrilateral $ABCD$ . The common points of segments $AK$ , $BL$ , $CM$ , $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?