Geometrically $C_1$ transfer principle for algebraically maximal valued fields

Let $(K,v)$ be an algebraically maximal valued field with divisible value group and perfect residue field $k$. A field is geometrically $C_1$ if every smooth projective separably rationally connected variety over it has a rational point. Geometrically $C_1$ transfer principle.

$$K\text{ is geometrically }C_1\iff k\text{ is geometrically }C_1.$$

The paper proposes this as a transfer principle relating a valued field to its residue field. It is presented as a conjectural principle and no resolution is supplied in the provided text.