The Sense Expansion Quest is a quest added as part of the New Years' Update. Yun, Myu, Sei, and Taku undertook the quest in Volume 11, starting in Chapter 1 and continuing in later chapters throughout that volume.
After speaking to the Priest NPC, the quest can be accepted. It costs 50SP (which are refunded if the quest is failed) to start. Then, three randomly generated quest stages are given. One is a "gathering" trial, one is an "aiding" trial, and one is a "subjugation" trial. Upon completion, an additional Sense slot is gained, as well as a free Sense.
Upon starting the quest, Yun, Myu, Sei, and Taku were given the following quest stages: Collect an 【Unfertilized Wyvern Egg】 from Wyvern Mountains; deliver 【Cure for Epidemy】 to the town ahead of the Highlands Area and save the sick; and subjugate the 【Emperor Foolegobug】.