3 problems

The all-values existence conjecture. For fixed $m$, holomorphic solutions with constant curvature in $G(m,n)$ can be constructed for every integer $r$ satisfying

The maximal-power conjecture. The maximal value of $r$ for which a holomorphic constant-curvature solution of $G(m,n)$ exists is

The Veronese minimal-curvature conjecture. The Veronese curves give rise to the smallest possible curvatures among holomorphic solutions in $G(m,n)$ with constant curvature.