Global convergence to a fixed point of the SUCPA-map

Let the SUCPA-map be $\mathbf{f}=[f_1,\ldots,f_K]:\mathbb{R}^K\to\mathbb{R}^K$, and let $\omega(\boldsymbol{\beta}^{[0]})$ denote the $\omega$-limit set of an initial condition $\boldsymbol{\beta}^{[0]}$. A point $\boldsymbol{\beta}^*$ is a fixed point when $\mathbf{f}(\boldsymbol{\beta}^*)=\boldsymbol{\beta}^*$. Global convergence conjecture. For each initial condition $\boldsymbol{\beta}^{[0]}\in\mathbb{R}^K$, there exists a unique fixed point $\boldsymbol{\beta}^*\in\mathbb{R}^K$, possibly depending on $\boldsymbol{\beta}^{[0]}$, such that $\omega(\boldsymbol{\beta}^{[0]})=\{\boldsymbol{\beta}^*\}$. The claim concerns convergence of every SUCPA iteration to a single fixed point; its resolution is not established by the supplied text.