Monotonicity conjecture for the $p$-normed flow index

Let $G$ be a bridgeless graph and let $d\ge 2$. The quantity $\phi_{d,p}(G)$ is the $d$-dimensional $p$-normed flow index. Monotonicity conjecture. As a function of $p$, $\phi_{d,p}(G)$ is continuous, increasing on $[1,2]$, and decreasing on $[2,\infty]$. Continuity is reported as proved in Rieg's Master's thesis, but the stated monotonicity remains conjectural.