Two disjoint shortest paths in the equal-hop path construction

Let $k$ be a node that is neither the source nor a virtual source. Let $i_{0}$, $i_{1}$, $j_{0}$, and $j_{1}$ be the quantities defined in Equations and. Let $c_{0}$ and $c_{1}$ be the two colors different from the color of $k$, and suppose that

$$i_{1}-i_{0}=j_{1}-j_{0}.$$

Two-path conjecture. Node $k$ always has two disjoint shortest paths $p_{0}$ and $p_{1}$ satisfying one of the following alternatives: (1) $p_{0}$ runs from $X_{0}$ through the color-$c_{0}$ neighbors to $k$, while $p_{1}$ runs from $X_{1}$ through the color-$c_{1}$ neighbors to $k$; or (2) $p_{0}$ runs from $X_{0}$ through the color-$c_{1}$ neighbors to $k$, while $p_{1}$ runs from $X_{1}$ through the color-$c_{0}$ neighbors to $k$. Here, an $h$-hop path set is a set of paths all having length $h$, so the assertion concerns shortest paths in the colored network construction. The claim is a structural property used to establish the relationship between the network equations and the existence of suitable disjoint paths; its resolution is not specified in the supplied text.