The unimodality conjecture for multidimensional flow numbers

For a fixed dimension $d\geq1$ and graph $G$, let $\Phi_d^p(G)$ be the multidimensional flow number for the $p$-norm, with $p\in[1,\infty]$.

Unimodality conjecture for flow numbers. As a function of $p$, $\Phi_d^p(G)$ is unimodal: there exists $p^*\in[1,\infty]$ such that it is non-decreasing for $p\leq p^*$ and non-increasing for $p\geq p^*$. Moreover,

$$p^*=2.$$

This conjecture proposes a precise dependence of multidimensional flow numbers on the norm parameter and identifies the Euclidean norm as the peak. The source says that the Euclidean case remains largely open.