Euler characteristics of cominuscule quantum K-theory

We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum K-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce that the sum of the structure constants associated to any product of Schubert classes is equal to 1. Equivalently, the sheaf Euler characteristic map extends to a ring homomorphism defined on the quantum K-theory ring.