Contraction of the restricted fixed-point map for the equilibrium prior

Prove that the modified fixed-point map \invbreve{F}(\chi), defined for the case \mathbb{K}_A \ne \mathbb{K}_B by setting Z_A = \sum_{k \in \mathbb{K}_{AB}} p_k(A), Z_B = \sum_{k \in \mathbb{K}_{AB}} p_k(B), \chi = \sqrt{Z_A/Z_B}\,\log P - \sqrt{Z_B/Z_A}\,\log(1-P), and \invbreve{F}(\chi) = (1/\sqrt{Z_A Z_B})\,\big(\Delta \invbreve{H} + \sum_{k\in\mathbb{K}_{AB}}(p_k(A)-p_k(B))\,\log\big(P(\chi)\,p_k(A) + (1-P(\chi))\,p_k(B)\big)\big) with \Delta \invbreve{H} = \invbreve{H}_A - \invbreve{H}_B and \invbreve{H}_\theta = -\sum_{k\in\mathbb{K}_{AB}} p_k(\theta)\,\log p_k(\theta), is a contraction on \mathbb{R}, i.e., that there exists \invbreve{q} < 1 such that \sup_{\chi\in\mathbb{R}} |\invbreve{F}'(\chi)| \le \invbreve{q}.

Background

In the Bayesian game BGame{N, K_A, K_B, M}, the equilibrium prior P^* is characterized as the unique minimizer of a convex objective, and several iterative methods are proposed to approximate P^*. When the feasible k-sets differ (\mathbb{K}_A \ne \mathbb{K}_B), the authors introduce a change of variables and a modified fixed-point map \invbreve{F}(\chi) intended to preserve contractivity on the log-odds-like coordinate \chi.

Establishing that \invbreve{F} is a contraction would yield a simple, globally convergent and error-controlled fixed-point iteration for computing the equilibrium prior in the asymmetric-support case, complementing the contraction result known for the symmetric case (\mathbb{K}_A = \mathbb{K}_B). The appendix contains a partial proof attempt; however, a complete proof is left open.