Uniqueness of the fixed-point line for the SUCPA-map

Let $\mathbf{f}=[f_1,\ldots,f_K]:\mathbb{R}^K\to\mathbb{R}^K$ be the SUCPA-map. For a fixed point $\boldsymbol{\beta}^*$ of $\mathbf{f}$, let $\boldsymbol{\lambda}=[\lambda,\ldots,\lambda]\in\mathbb{R}^K$ and define the straight line $\mathcal{S}(\boldsymbol{\beta}^*)=\{\boldsymbol{\beta}=\boldsymbol{\lambda}+\boldsymbol{\beta}^*\}$. Uniqueness conjecture for the fixed-point line. The map $\mathbf{f}$ has a unique straight line of fixed points of the form $\mathcal{S}(\boldsymbol{\beta}^*)$. The translation invariance described before the conjecture shows that fixed points occur along lines in the all-ones direction. The supplied status evidence says that the case $K=2$ is formally proven, while the general claim is supported by numerical evidence and remains unresolved.