The uniform-personalization conjecture for optimal PageRank link structures

Let $\mathcal{E}_\mathrm{in}(\mathcal{I})\neq\emptyset$ and $\mathcal{E}_{\overline{\mathcal{I}}}$ be given. Let $\mathcal{E}_{\mathcal{I}}$ and $\mathcal{E}_\mathrm{out}(\mathcal{I})$ be such that $\boldsymbol{\pi}^T\boldsymbol{e}_{\mathcal{I}}$ is maximal under Assumption A. If $\boldsymbol{z}=\frac{1}{n}\boldsymbol{1}$, then there exists $j\in\overline{\mathcal{I}}$ such that $(j,i)\in\mathcal{E}_\mathrm{in}(\mathcal{I})$, where $i\in\operatorname*{argmax}_{k}\boldsymbol{v}_k$. Uniform-personalization conjecture. Under these conditions, the first node of $\mathcal{I}$ in the forward chain of an optimal link structure is necessarily a child of some node of $\overline{\mathcal{I}}$. This conjecture concerns the structure of PageRank-maximizing websites and whether uniform personalization rules out the exceptional configuration found for nonuniform personalization.