Buch–Mihalcea rationality conjecture for Gromov–Witten varieties

Let $X$ be a cominuscule variety, and let $d_{\rm \max}$ denote the minimal integer such that a curve of degree $d_{\rm \max}$ passes through any two points of $X$. For $d>d_{\rm \max}$ and general points $x_1,x_2,x_3\in X$, let $GW_d(x_1,x_2,x_3)$ be the Gromov–Witten variety of degree-$d$ curves passing through these points. Buch–Mihalcea conjecture. The variety $GW_d(x_1,x_2,x_3)$ is rational. This rationality is the geometric input for the quantum-to-classical principle in equivariant quantum $K$-theory; the paper proves the conjecture in several cases but presents the general statement as conditional.