Computable fixed-point selection for nonexpansive mappings on two-dimensional Banach spaces

Let $E$ be a uniformly convex, smooth, computable Banach space of dimension two, and let $K\subseteq E$ be bounded, convex, and located, meaning that its distance function is computable. Write $\mathcal{K}^{\operatorname{co}}(K)\setminus\{\emptyset\}$ for the represented space of nonempty co-semi-decidable convex subsets of $K$, and $\operatorname{\mathcal{N}}(K)$ for the relevant space of nonexpansive self-mappings of $K$. Computable fixed-point selection conjecture. The multi-valued mapping

$$\operatorname{Fix}^{-1}\colon \mathcal{K}^{\operatorname{co}}(K)\setminus\{\emptyset\}\rightrightarrows \operatorname{\mathcal{N}}(K)$$

is computable. This is presented as a possible generalisation of the preceding two-dimensional Hilbert-space result to uniformly convex and smooth real Banach spaces. The source gives no resolution evidence for this claim, so its status remains open.