Singleton reverse-Funk Busemann points vs. extreme rays of Csc when C ⊂ Csc

Determine whether, in a complete order unit space (V, C, u) with state space S where Csc denotes the cone of bounded affine functions on S that can be written as a difference of two nonnegative affine upper semi-continuous functions, the singleton reverse-Funk Busemann points h(x) = log sup_{φ∈S} g(φ)/φ(x) (with g a w*-upper semi-continuous, nonnegative, affine function on S of supremum 1) correspond exactly to the extreme rays {λg : λ > 0} of Csc in the case C is strictly contained in Csc (i.e., C ⊂ Csc).

Background

The paper characterizes reverse-Funk Busemann points for cones in complete order unit spaces and, under the assumption C = Csc, identifies the singleton reverse-Funk Busemann points precisely with normalized extreme vectors of the cone. Here, Csc is the cone of bounded affine functions on the state space S representable as differences of two nonnegative affine upper semi-continuous functions.

However, when the original cone C is a proper subset of Csc (C ⊂ Csc), the authors do not know whether the same correspondence holds. Resolving this would extend the structural link between horofunction geometry (singleton reverse-Funk points) and the order-theoretic extremal structure (extreme rays of Csc) beyond the reflexive/normal state setting.