The finite-map criterion for properness

Let $f$ be a map of finite topological spaces. Say that a fibre has two points bounded above but not below when its two points have a common upper bound but no common lower bound, and say that two points are topologically indistinguishable when they have the same open neighbourhoods. The finite-map properness conjecture. The class $\{f\}^{lr}$ is the class of all proper maps if and only if $f$ is closed, not surjective, and satisfies all of the following: $f$ has a fibre with two points bounded above but not below; $f$ has a fibre with two distinct topologically indistinguishable points; and the image of $f$ is not both open and closed. This is a proposed classification of the finite maps whose double orthogonal is exactly the class of proper maps; the source gives no proof or resolution.