The non-Hausdorff Brouwer fixed point conjecture

Let $K$ be a topological space equipped with maps $K\to\{\bullet\}$ lying in both indicated right-orthogonal classes: the class generated by the finite-space maps $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$, and the double orthogonal of the maps between finite spaces of size less than $5$ used to characterize proper maps. The non-Hausdorff Brouwer fixed point conjecture. Every endomorphism of $K$ has a fixed point. The assertion extends the Brouwer fixed point theorem beyond Hausdorff spaces; the paper proves it for separable metric spaces but leaves the general case open.