This page explains how to start and complete the The Gut Check Challenge shrine quest.
How to Start the Quest
To start the quest, you need to go to Gut Check Rock, which is in the area northwest of East Akkala Stable, which is near Katosa Aug Shrine. If you have unlocked Zuna Kai Shrine, you can go west from there to find it. Then talk to Bayge at Gut Check Rock to start the quest.
How to Complete the Quest
Check the Gorae Torr Shrine page for details on completing the quest.