FKMO 2020 Problem 2

There are $2020$ groups, each of which consists of a boy and a girl, such that each student is contained in exactly one group. Suppose that the students shook hands so that the following conditions are satisfied:

• boys didn't shake hands with boys, and girls didn't shake hands with girls;
• in each group, the boy and girl had shake hands exactly once;
• any boy and girl, each in different groups, didn't shake hands more than once;
• for every four students in two different groups, there are at least three handshakes.

Prove that one can pick $4038$ students and arrange them at a circular table so that every two adjacent students had shake hands.