The μ-PageRank monotonicity and range conjecture

Let $G$ be a graph, let $[?]$-PageRank be the PageRank vector $[?]$ of $G$, and write $[?](i)$ for the value at node $i$. For $[?]>0$, let $[?]$ denote an increment of the parameter.

μ-PageRank monotonicity and range conjecture. The function $([?])_i$ is monotonic for all $i$ and

$$\max_i\pi_\mu(i)-\min_i\pi_\mu(i)\leq\max_i\pi_{\mu+\alpha}(i)-\min_i\pi_{\mu+\alpha}(i)$$

for all $[?]>0$.

This is the formal version of the two preceding empirical claims. The source motivates it with numerical experiments, but gives no proof or resolution.