The generalized $S^2_p$-flow conjecture

For a bridgeless graph $G$, let

$$S_p^{d-1}:=\{\boldsymbol{x}\in\mathbb{R}^d:\lVert\boldsymbol{x}\rVert_p=1\}$$

be the unit $p$-sphere. The generalized $S^2_p$-flow conjecture. For every bridgeless graph $G$ and every $p\in[1,\infty]$,

$$\phi_{3,p}(G)=2.$$

This extends Jain's $S^2$-flow conjecture from the Euclidean sphere to all $p$-norms. The source notes that the case $p=\infty$ is proved, while the general assertion is suggested using the preceding monotonicity and duality conjectures.