Duality conjecture for normed flow indices

Let $G$ be a bridgeless graph and let $d\ge 2$ be an integer. For $p\in[1,2]$ and $q\in[2,\infty]$, write $\phi_{d,p}(G)$ and $\phi_{d,q}(G)$ for the corresponding flow indices. Duality conjecture for normed flow indices. For every $p\in[1,2]$, there exists $q\in[2,\infty]$ such that

$$\phi_{d,p}(G)=\phi_{d,q}(G),$$

and conversely, for every $q\in[2,\infty]$, there exists $p\in[1,2]$ satisfying the same equality. The source contrasts this with the usual duality relation $1/p+1/q=1$ and leaves the asserted existence open.