JBMO-SL 2020 C2

Viktor and Natalia bought 2020 buckets of ice-cream and want to organize a degustation schedule with 2020 rounds such that:

• In every round, each one of them tries 1 ice-cream, and those 2 ice-creams tried in a single round are different from each other.
• At the end of the 2020 rounds, each one of them has tried each ice-cream exactly once.

We will call a degustation schedule fair if the number of ice-creams that were tried by Viktor before Natalia is equal to the number of ice creams tried by Natalia before Viktor.

Prove that the number of fair schedules is strictly larger than $2020!\left(2^{1010}+(1010!)^{2}\right)$.