Projection invariance of fixed points for sunny nonexpansive retractions

Let $E$ be a smooth and uniformly convex Banach space of dimension two. Let $K\subseteq E$ be nonempty, closed, bounded, and convex, and let $f\colon K\to E$ be nonexpansive with $\operatorname{Fix}(f)\neq\emptyset$. Let $P\colon E\to K$ be the sunny nonexpansive retraction onto $K$. Projection invariance conjecture. One has

$$\operatorname{Fix}(P\circ f)=\operatorname{Fix}(f).$$

The claim would justify replacing a Hilbert-space projection by a sunny nonexpansive retraction in the fixed-point argument. The source presents it as an unresolved question and supplies no proof or disproof, so it remains open.