The $(2k-1)$-edge-connected bounded-out-degree orientation conjecture

Let $G$ be a graph, let $k$ be an integer with $k\ge 3$, and let $p:V(G)\rightarrow Z_k$ satisfy

$$|E(G)| \stackrel{k}{\equiv} \sum_{v\in V(G)}p(v).$$

A $p$-orientation is an orientation satisfying the prescribed modulo-$k$ out-degree conditions encoded by $p$. The $(2k-1)$-edge-connected bounded-out-degree conjecture. If $G$ is $(2k-1)$-edge-connected, then $G$ has a $p$-orientation such that, for every vertex $v$,

$$\left\lfloor\frac{d_G(v)}{2}\right\rfloor-(k-1)\le d_G^+(v)\le \left\lceil\frac{d_G(v)}{2}\right\rceil+(k-1).$$

Furthermore, for an arbitrary vertex $z_0$, $d_G^+(z_0)$ can be assigned any plausible integer value in this interval. This would improve the edge-connectivity required by the preceding result from $2k$ to $2k-1$; the conjecture is presented as qualitative and remains unresolved in the supplied text.