Mild boundedness and grouplessness equivalence conjecture

Let $k$ be an algebraically closed field of characteristic zero, and let $X$ be a projective variety over $k$. Consider the properties: $X$ is mildly bounded; $X$ has no rational curves; for every curve $C$, there is an integer $d\geq 1$ such that $\operatorname{Hom}^{\geq d}(C,X)$ is zero-dimensional; and $X$ is groupless.

Mild boundedness and grouplessness conjecture. In this situation, mild boundedness is equivalent to having no rational curves, and the condition that $\operatorname{Hom}^{\geq d}(C,X)$ is zero-dimensional for every curve $C$ is equivalent to grouplessness.

The source proves several one-way implications, including mild boundedness implying no rational curves and the Hom-scheme condition implying grouplessness; over uncountable fields it also proves the converse from the Hom-scheme condition to mild boundedness. The asserted equivalences remain conjectural.