Geometric $C_1$ transfer 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 called geometrically $C_1$ if every smooth projective separably rationally connected variety over it has a rational point. Geometric $C_1$ transfer conjecture.

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

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