Prove positivity of the explicit classical-fixed-point count for compressive domains

Prove directly that the explicit count of classical fixed points in a compressive domain, given by 1 plus the sum of the surplus multiplicities of its inward directions minus the degree times one less than the number of boundary points, is positive.

Background

For a compressive domain U with boundary points x₁,…,xₙ and inward directions v₁,…,vₙ, Proposition 2.1 expresses the number of classical fixed points in U as 1 + Σᵢ s_f(vᵢ) − d(n−1), where s_f(vᵢ) denotes the surplus multiplicity and d is the degree of the rational function. The authors note that a direct proof of Theorem C would ideally establish positivity of this expression. They were unable to make progress on that approach, although they subsequently prove Theorem C by a different argument.