The algebraically closed field-topology independence conjecture

Let $C$ be an algebraic curve over an algebraically closed field $K$, and let $\tau_1$ and $\tau_2$ be independent field topologies on $K$. Two topologies on a set are independent if every nonempty open set for one intersects every nonempty open set for the other.

Curve-topology independence conjecture. The topologies $\tau_1$ and $\tau_2$ induce independent topologies on $C(K)$.

This is a weaker variant of the preceding proposed technical statement, obtained by assuming that the field is algebraically closed and omitting t-henselianity. The source gives no resolution.