This ability will only trigger if the player controls more lands at the beginning of upkeep. This condition is checked again at the start of resolution, and if not true, then nothing happens.

Rivalry : règles, fonctionnement et interactions (MTG)

Rivalry is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

At the beginning of each player's upkeep, if that player controls more lands than each other player, this enchantment deals 2 damage to them.

How it works : Rivalry

Original English text — Wizards of the Coast / Scryfall

#1October 4, 2004· Wizards of the Coast
This ability will only trigger if the player controls more lands at the beginning of upkeep. This condition is checked again at the start of resolution, and if not true, then nothing happens.
No other card is named in this ruling.